This document discusses simplifying variable expressions. It begins with an essential question about adding, subtracting, multiplying, and dividing variable expressions. It then provides the vocabulary for order of operations, including grouping symbols, exponents, multiplication, division, addition, and subtraction. Examples are provided to demonstrate simplifying expressions using order of operations. The examples include solving for variables and writing expressions in terms of variables.
The document summarizes key concepts about composition functions:
1. Composition functions are not commutative.
2. They are associative.
3. Identity functions leave other functions unchanged when composed.
The document also provides examples of determining the component functions of composition functions and finding inverse functions.
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
(1) The document defines four functions: f(x)=2x-6, g(x)=-3x+5, h(x)=x^2-1, k(x)=(2x+5)^2-1. It then defines operations on functions such as addition, subtraction, multiplication, and composition.
(2) Examples are given of calculating the sum, difference, and product of two functions, as well as the composite function gโf. The domain and range of the composite functions are discussed.
(3) The inverse of a function is defined. Examples inverse functions are calculated from relations provided in the text.
This document contains an exercise set from a chapter on functions. It includes 35 multi-part math problems testing concepts like domains and ranges of functions, rates of change, and word problems involving temperature, speed, and geometric shapes. The problems cover skills like determining maximum/minimum values, solving equations, and sketching graphs of functions.
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
The document discusses inverse functions. It defines a one-to-one function as a function where the horizontal line test shows that every horizontal line intersects the graph at most one point. This ensures that each input is mapped to a single output. An inverse function undoes the original function - if f(x) is the original function, its inverse f^-1(x) satisfies f^-1(f(x)) = x.
This document discusses surds, indices, and logarithms. It begins by defining radicals, surds, and irrational numbers. Some general rules for operations with surds like multiplication, division, and simplification are provided. The document then covers rules and operations for indices like exponentiation, roots, and properties like distributing exponents. Examples are given to demonstrate applying these index rules. The document concludes by defining logarithms as the inverse of exponentiation and provides an example equation.
This document discusses simplifying variable expressions. It begins with an essential question about adding, subtracting, multiplying, and dividing variable expressions. It then provides the vocabulary for order of operations, including grouping symbols, exponents, multiplication, division, addition, and subtraction. Examples are provided to demonstrate simplifying expressions using order of operations. The examples include solving for variables and writing expressions in terms of variables.
The document summarizes key concepts about composition functions:
1. Composition functions are not commutative.
2. They are associative.
3. Identity functions leave other functions unchanged when composed.
The document also provides examples of determining the component functions of composition functions and finding inverse functions.
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
(1) The document defines four functions: f(x)=2x-6, g(x)=-3x+5, h(x)=x^2-1, k(x)=(2x+5)^2-1. It then defines operations on functions such as addition, subtraction, multiplication, and composition.
(2) Examples are given of calculating the sum, difference, and product of two functions, as well as the composite function gโf. The domain and range of the composite functions are discussed.
(3) The inverse of a function is defined. Examples inverse functions are calculated from relations provided in the text.
This document contains an exercise set from a chapter on functions. It includes 35 multi-part math problems testing concepts like domains and ranges of functions, rates of change, and word problems involving temperature, speed, and geometric shapes. The problems cover skills like determining maximum/minimum values, solving equations, and sketching graphs of functions.
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
The document discusses inverse functions. It defines a one-to-one function as a function where the horizontal line test shows that every horizontal line intersects the graph at most one point. This ensures that each input is mapped to a single output. An inverse function undoes the original function - if f(x) is the original function, its inverse f^-1(x) satisfies f^-1(f(x)) = x.
This document discusses surds, indices, and logarithms. It begins by defining radicals, surds, and irrational numbers. Some general rules for operations with surds like multiplication, division, and simplification are provided. The document then covers rules and operations for indices like exponentiation, roots, and properties like distributing exponents. Examples are given to demonstrate applying these index rules. The document concludes by defining logarithms as the inverse of exponentiation and provides an example equation.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
This document discusses partial differentiation in mathematics. It provides examples of taking the partial derivative of functions z with respect to x and y. The examples include functions such as z=x^2 + xy + y^2, z=(2x-y)(x+3y), z=sin(3x+2y), and z=tan(3x+4y). Formulas are given for calculating the partial derivatives using multiplication and division rules of functions. Exercises are provided to have the reader calculate the partial derivatives of additional functions.
1. The document contains examples of evaluating limits as the variable approaches certain values.
2. Several limits were found to be indeterminate forms that require further algebraic manipulation to find the limit.
3. Key observations were made about the behavior of functions as the variable approaches values like noticing a function approaches a certain value as the variable nears another value.
1. The document defines several functions and their domains and ranges. It also defines function compositions.
2. An example function is defined as f(x) = 2x and another is defined as g(x) = x + 1. It is shown that these functions are equal.
3. Several other example functions are defined, including trigonometric, polynomial, and rational functions.
4. Function compositions are defined for specific functions f and g over the domain of positive integers, and examples are given to illustrate function composition.
This document contains a summary of the key points to address in answering additional mathematics questions from a 2015 SPM exam. It includes 24 questions covering topics like functions, vectors, probability, calculus, and statistics. For each question, it lists the correct answer or working, identified with a check mark. It provides guidance on acceptable versus unacceptable responses. The summary aims to help students understand the essential information and logical reasoning required to solve additional math problems.
The document is the cover page of a mathematics question paper containing instructions and details about the exam. It states that the exam is for 2 1/2 hours with a maximum of 100 marks. The paper contains four sections. It instructs students to check for fairness of printing and inform the supervisor if any issues are found.
This document contains solutions to exercises from a chapter on partial derivatives. It includes:
1) Solutions to 14 sets of partial derivative exercises involving functions of two or more variables.
2) Discussion of limits of functions as the variables approach certain points, including cases where the limit does not exist.
3) Graphical representations of functions of two variables and their level curves.
The document provides detailed worked solutions to multiple partial derivative practice problems across several pages.
This document contains exercises related to inverse functions and their properties. It includes 53 multi-part exercises involving determining if functions are inverses, composing functions, finding inverse functions, and evaluating derivatives of inverse functions. The exercises involve algebraic manipulation and graphical analysis of functions and their inverses.
Solution Manual : Chapter - 02 Limits and ContinuityHareem Aslam
ย
This document contains solutions to exercise sets on limits and continuity from a textbook. Exercise Set 2.1 contains solutions to 23 problems evaluating limits of functions as the input values approach certain numbers. Exercise Set 2.2 contains solutions to 38 similar problems evaluating limits. Exercise Set 2.3 contains solutions to 23 additional limit evaluation problems. The document provides the step-by-step workings and conclusions for each problem.
This document provides information and examples on multiplying polynomials, including:
1) Multiplying a monomial and polynomial using the distributive property.
2) Multiplying two polynomials using both the horizontal and vertical methods.
3) Factoring trinomials and identifying similar and conjugate binomials. Methods like FOIL and grouping are discussed.
This module introduces linear functions of the form f(x) = mx + b. Learners will develop skills to determine the slope, trend, intercepts, and points of linear functions. The module is designed to help learners:
1) Determine slope, trend, intercepts, and points of a linear function given f(x) = mx + b
2) Determine f(x) = mx + b given various conditions like slope and intercepts, two points, etc.
The document provides lessons that explain how to find these properties of linear functions through examples. Practice problems are also included for learners to test their understanding.
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
This document contains notes from Chapter 2 on rational numbers and probability. Key concepts covered include: adding, subtracting, multiplying and dividing rational numbers; properties of numbers like commutative, associative, identity, and inverse; theoretical and experimental probability; and probability of compound events being dependent or independent. Examples are provided to illustrate concepts like finding probabilities of drawing different colored marbles from a bag without and with replacement.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
The document defines and provides examples of several types of functions including:
1) Constant functions where f(x) = a for all values of x.
2) Linear functions of the form f(x) = ax + b.
3) Quadratic functions of the form f(x) = ax2 + bx + c.
4) Polynomial functions which are the sum of terms with variables raised to various powers.
This document provides instruction on factoring perfect square trinomials and the difference of squares. It includes links to video explanations of factoring perfect square trinomials. It provides examples of factoring a perfect square trinomial and the difference of squares, including splitting the perfect square and finding the difference. Students are instructed to practice these skills on interactive math websites and problems, and to complete an assessment with multiple choice questions in an online factoring Gizmo simulation.
The document provides examples of finding the zeros of quadratic functions by factorizing and setting each factor equal to 0. It then lists 5 additional quadratic functions and assigns the reader to find their zeros.
The technical report presents two social recommendation methods that incorporate semantics from tags: a user-based semantic collaborative filtering and an item-based semantic collaborative filtering. The methods aim to find semantically similar users/items and recommend relevant social items. Experimental results show the methods improve recommendation quality and address issues like polysemy, synonymy, and semantic interoperability compared to methods without semantics.
The document provides definitions and properties of circles. It defines key terms like radius, diameter, chord, arc, and central angle. It then works through an example problem to find the measures of angles in a quadrilateral using properties of angles in circles. Specifically, it uses the fact that opposite angles of any quadrilateral inscribed in a circle are equal and sums of angles of a quadrilateral add to 360 degrees.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
This document discusses partial differentiation in mathematics. It provides examples of taking the partial derivative of functions z with respect to x and y. The examples include functions such as z=x^2 + xy + y^2, z=(2x-y)(x+3y), z=sin(3x+2y), and z=tan(3x+4y). Formulas are given for calculating the partial derivatives using multiplication and division rules of functions. Exercises are provided to have the reader calculate the partial derivatives of additional functions.
1. The document contains examples of evaluating limits as the variable approaches certain values.
2. Several limits were found to be indeterminate forms that require further algebraic manipulation to find the limit.
3. Key observations were made about the behavior of functions as the variable approaches values like noticing a function approaches a certain value as the variable nears another value.
1. The document defines several functions and their domains and ranges. It also defines function compositions.
2. An example function is defined as f(x) = 2x and another is defined as g(x) = x + 1. It is shown that these functions are equal.
3. Several other example functions are defined, including trigonometric, polynomial, and rational functions.
4. Function compositions are defined for specific functions f and g over the domain of positive integers, and examples are given to illustrate function composition.
This document contains a summary of the key points to address in answering additional mathematics questions from a 2015 SPM exam. It includes 24 questions covering topics like functions, vectors, probability, calculus, and statistics. For each question, it lists the correct answer or working, identified with a check mark. It provides guidance on acceptable versus unacceptable responses. The summary aims to help students understand the essential information and logical reasoning required to solve additional math problems.
The document is the cover page of a mathematics question paper containing instructions and details about the exam. It states that the exam is for 2 1/2 hours with a maximum of 100 marks. The paper contains four sections. It instructs students to check for fairness of printing and inform the supervisor if any issues are found.
This document contains solutions to exercises from a chapter on partial derivatives. It includes:
1) Solutions to 14 sets of partial derivative exercises involving functions of two or more variables.
2) Discussion of limits of functions as the variables approach certain points, including cases where the limit does not exist.
3) Graphical representations of functions of two variables and their level curves.
The document provides detailed worked solutions to multiple partial derivative practice problems across several pages.
This document contains exercises related to inverse functions and their properties. It includes 53 multi-part exercises involving determining if functions are inverses, composing functions, finding inverse functions, and evaluating derivatives of inverse functions. The exercises involve algebraic manipulation and graphical analysis of functions and their inverses.
Solution Manual : Chapter - 02 Limits and ContinuityHareem Aslam
ย
This document contains solutions to exercise sets on limits and continuity from a textbook. Exercise Set 2.1 contains solutions to 23 problems evaluating limits of functions as the input values approach certain numbers. Exercise Set 2.2 contains solutions to 38 similar problems evaluating limits. Exercise Set 2.3 contains solutions to 23 additional limit evaluation problems. The document provides the step-by-step workings and conclusions for each problem.
This document provides information and examples on multiplying polynomials, including:
1) Multiplying a monomial and polynomial using the distributive property.
2) Multiplying two polynomials using both the horizontal and vertical methods.
3) Factoring trinomials and identifying similar and conjugate binomials. Methods like FOIL and grouping are discussed.
This module introduces linear functions of the form f(x) = mx + b. Learners will develop skills to determine the slope, trend, intercepts, and points of linear functions. The module is designed to help learners:
1) Determine slope, trend, intercepts, and points of a linear function given f(x) = mx + b
2) Determine f(x) = mx + b given various conditions like slope and intercepts, two points, etc.
The document provides lessons that explain how to find these properties of linear functions through examples. Practice problems are also included for learners to test their understanding.
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
This document contains notes from Chapter 2 on rational numbers and probability. Key concepts covered include: adding, subtracting, multiplying and dividing rational numbers; properties of numbers like commutative, associative, identity, and inverse; theoretical and experimental probability; and probability of compound events being dependent or independent. Examples are provided to illustrate concepts like finding probabilities of drawing different colored marbles from a bag without and with replacement.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
The document defines and provides examples of several types of functions including:
1) Constant functions where f(x) = a for all values of x.
2) Linear functions of the form f(x) = ax + b.
3) Quadratic functions of the form f(x) = ax2 + bx + c.
4) Polynomial functions which are the sum of terms with variables raised to various powers.
This document provides instruction on factoring perfect square trinomials and the difference of squares. It includes links to video explanations of factoring perfect square trinomials. It provides examples of factoring a perfect square trinomial and the difference of squares, including splitting the perfect square and finding the difference. Students are instructed to practice these skills on interactive math websites and problems, and to complete an assessment with multiple choice questions in an online factoring Gizmo simulation.
The document provides examples of finding the zeros of quadratic functions by factorizing and setting each factor equal to 0. It then lists 5 additional quadratic functions and assigns the reader to find their zeros.
The technical report presents two social recommendation methods that incorporate semantics from tags: a user-based semantic collaborative filtering and an item-based semantic collaborative filtering. The methods aim to find semantically similar users/items and recommend relevant social items. Experimental results show the methods improve recommendation quality and address issues like polysemy, synonymy, and semantic interoperability compared to methods without semantics.
The document provides definitions and properties of circles. It defines key terms like radius, diameter, chord, arc, and central angle. It then works through an example problem to find the measures of angles in a quadrilateral using properties of angles in circles. Specifically, it uses the fact that opposite angles of any quadrilateral inscribed in a circle are equal and sums of angles of a quadrilateral add to 360 degrees.
This document provides information about congruent triangles, including definitions of key terms like congruent triangles, side-side-side (SSS) postulate, side-angle-side (SAS) postulate, and angle-side-angle (ASA) postulate. It includes examples of using these postulates to determine if triangles are congruent. The document also contains activities where students draw triangles to explore congruence. Finally, it provides multi-step word problems involving finding missing side lengths and angle measures of congruent triangles.
1) The document provides guidance on how to evaluate whether websites contain reliable information for research purposes. It discusses checking who authored the content and whether the site achieves its stated purpose.
2) The document recommends several websites that can help determine if a site is reliable, as well as tips for evaluating a site's relevance and sorting reliable resources.
3) It concludes by encouraging the reader to test their skills in evaluating websites and thanks sources that helped create the document.
Whats the big idea with social media media140-2012Kate Carruthers
ย
Talk given at Media140 Perth 2012
More detailed notes on the talk here:
http://katecarruthers.com/blog/2012/05/whats-the-big-idea-with-social-media-media140/
The keynote is the teaching material for the UOID + AHMI course in 2013. It is an multidisciplinary course for the cooperation between NTUST design and NTU IT students. The course is held on NTUST. The purpose of the course is creating assisting or supportive APPS that are needed and appropriate for underprivileged people in Taiwan. The lectures are drhhtang and Mike Chen. The content of the slide is describing the process of human-centered design process and the design brief for 2013.
Here are the steps to solve problem #1 on page 74:
1) Simplify the expression: -3(x - 5)
2) Use the property that anything inside the parentheses will be opposite if there is a negative sign outside: -3(x - 5) = -3x + 15
3) Simplify: -3x + 15
The simplified expression is: -3x + 15
1. The document reviews multiplying polynomials including binomials and trinomials using the FOIL method. It provides examples of multiplying binomial expressions.
2. It then has students practice simplifying the addition and subtraction of polynomials and multiplying binomial expressions using FOIL.
3. The document concludes with examples of using polynomials to represent and calculate the area of a rectangle.
The document discusses multiplying and dividing variable expressions. It provides examples of simplifying expressions using the distributive property and the property of the opposite of a sum. It also demonstrates dividing variable expressions by writing the division as a fraction and simplifying. Key steps include distributing terms, dividing each term in the numerator by the denominator, and evaluating expressions for given variable values.
This document provides examples and explanations of operations involving polynomials and rational expressions. It covers factoring polynomials, evaluating polynomial expressions, adding, subtracting, multiplying and dividing rational expressions, and simplifying complex fractions and expressions with radicals. Step-by-step solutions are shown for problems such as factoring expressions, evaluating polynomials for given values, combining like rational expressions, rationalizing denominators, and more. The document demonstrates various techniques for working with polynomials and rational expressions.
This document provides examples and explanations of operations and concepts involving polynomial and rational expressions. It begins with examples of factoring polynomials and using the factored form to evaluate expressions. It then covers topics such as combining like terms in rational expressions, multiplying and dividing rational expressions using factoring, simplifying complex fractions, and rationalizing denominators involving radicals. The document aims to demonstrate techniques for working with polynomials and rational expressions through step-by-step examples and explanations of related concepts.
This document provides information about polynomial operations including:
1) Defining polynomials as algebraic expressions involving integer powers of a variable and real number coefficients.
2) Examples of adding, subtracting, and multiplying polynomials by using vertical or FOIL methods.
3) Important formulas for polynomial operations such as (a + b)(a - b) = a2 - b2 and (a + b)2 = a2 + 2ab + b2.
4) Worked examples of applying these formulas and methods to polynomials involving single and multiple variables.
The document discusses the distributive property and how it is used to simplify algebraic expressions. It provides examples of distributing terms over addition and subtraction, such as 5(x + 7) = 5x + 35. It also discusses like terms and how they can be combined when simplifying expressions.
This document discusses factoring the difference of two squares. It begins with examples of factoring binomials of the form (a+b)(a-b) where a and b are perfect squares. It then states the formula for factoring the difference of two squares as a2 - b2 = (a+b)(a-b). Several examples are worked out step-by-step applying this formula to factor expressions like x4 - 81, 4x2 - 36, and 25x2 - 36y2. The document concludes with a short quiz of 5 examples for the reader to try factoring the difference of two squares on their own.
Factoring 15.3 and 15.4 Grouping and Trial and Errorswartzje
ย
This document discusses various methods for factoring trinomials of the form Ax^2 + Bx + C. It begins by outlining three main methods: trial and error, factoring by grouping, and the box method. It then provides examples of using the factoring by grouping method, demonstrating how to find two numbers whose product is AC and sum is B. The document also covers special cases like factoring the difference of squares using the form A^2 - B^2 = (A-B)(A+B), and factoring perfect square trinomials using the form A^2 + 2AB + B^2 = (A+B)^2. In all, it thoroughly explains the step-by
1) The document contains 11 math word problems involving calculations with integers, variables, equations, and inequalities.
2) The problems cover topics like evaluating expressions, combining like terms, solving equations, and word problems about costs and sharing money.
3) The full solutions are shown for each problem.
The document is a quiz on order of operations and the distributive property with 4 multiple choice questions. It tests evaluating expressions using order of operations, identifying the order of operations in expressions, distributing terms in expressions, and simplifying expressions using the distributive property.
The students will learn to use the distributive property to simplify expressions by distributing terms being multiplied to terms inside parentheses. The distributive property distributes the number outside the parentheses to each term inside. Examples are provided to demonstrate distributing terms and combining like terms to simplify expressions.
This PowerPoint covers about 90% of the material that will be on the Chapter 8 test. It includes steps to solve linear equations for y in terms of x, naming the slope (m) and y-intercept (b), identifying ordered pairs, determining if a relation is a function, writing rules for linear functions from tables of values, and finding the slope.
The document provides objectives and examples for adding and subtracting polynomials. The objectives are to: 1) Add polynomials 2) Subtract polynomials 3) Solve problems involving adding and subtracting polynomials. Examples are provided to demonstrate representing quantities with tiles, adding polynomials by grouping like terms, and subtracting polynomials using the keep, change, change process.
The document provides examples of using distributive properties to simplify expressions, as well as examples of calculating area given length and width or vice versa. It contains practice problems for students to work through involving simplifying expressions using distributive properties and calculating area from given dimensions or unknown dimensions from a given area.
Re call basic operations in mathematics Nadeem Uddin
ย
The document discusses basic mathematical operations - addition, subtraction, multiplication and division. It provides examples and exercises for performing each operation on numbers and algebraic expressions. It also covers concepts like coefficients, bases and exponents of algebraic terms, polynomials, and the order of operations (BODMAS rule).
1. The document discusses adding and subtracting fractions with both equal and unequal denominators. It provides examples of finding a common denominator and then adding or subtracting the numerators.
2. It also provides examples of factoring expressions before combining like terms, as well as canceling terms before simplifying fractions.
3. The document concludes by solving two multi-step word problems involving fractions.
The document discusses the distributive property and how it is used to simplify algebraic expressions. It provides examples of distributing terms over addition and subtraction. It defines terms, coefficients, and like terms. It then gives examples of simplifying expressions by combining like terms.
Here are 3 practice problems from the problem set with solutions:
1) Simplify: 8x + 12x
20x
2) Evaluate the expression 5x + 2x when x = 3:
7x
21
3) Simplify and combine like terms: 4y - 2y + 7y - y
8y
Work through the rest of the assigned problems carefully and check your work. Ask for help if you get stuck on any part of the process. Tackling a full problem set is an excellent way to reinforce the concepts and build skills in working with variable expressions.
This document provides an outline of topics in algebra including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples and explanations for each topic.
The document defines and provides examples of angle relationships including adjacent angles, linear pairs, vertical angles, complementary angles, and supplementary angles. It defines each term and provides examples of identifying angle pairs that satisfy each relationship. It also includes examples of using properties of these angle relationships to solve problems, such as finding missing angle measures.
The document defines various terms related to angle measure including ray, angle, vertex, acute angle, obtuse angle, and angle bisector. It then provides examples measuring angles in a figure and solving an equation involving angle measures.
This document discusses finding points and midpoints on line segments. It defines midpoint as the point halfway between two endpoints and provides the formula to calculate it. Several examples are given to demonstrate how to find the midpoint of a segment, locate a point at a fractional distance from one endpoint, and find a point where the ratio of distances from the endpoints is a given ratio. The key concepts covered are calculating midpoints using averages of x- and y-coordinates, and setting up and solving equations to locate interior points using fractional or ratio distances along a segment.
This document defines key vocabulary terms related to line segments and distance. It defines a line segment as a portion of a line distinguished by endpoints, and defines betweenness of points and the term "between." It also defines congruent segments, constructions, distance, and irrational numbers. Several examples are provided to demonstrate calculating distances between points on a number line, using a ruler to measure segments, and applying the Pythagorean theorem.
This document defines key vocabulary terms related to line segments and distance, including line segment, betweenness of points, congruent segments, and distance formula. It provides examples of calculating distances between points on number lines and using the Pythagorean theorem to find distances between points graphed on a coordinate plane. Examples include measuring line segments with a ruler, finding distances by adding or subtracting measures, and applying the distance formula and Pythagorean theorem to solve for unknown distances.
This document introduces basic geometry concepts such as points, lines, planes, and their intersections. It defines a point as having no size or shape, a line as an infinite set of collinear points, and a plane as a flat surface extending indefinitely. Examples demonstrate identifying geometric shapes from real-world objects and graphing points and lines on a coordinate plane. The summary defines key terms and provides examples of geometric concepts and relationships.
The document discusses inverse functions and relations. It defines an inverse relation as one where the coordinates of a relation are switched, and an inverse function as one where the domain and range of a function are switched. It provides examples of finding the inverse of specific relations and functions by switching their coordinates or domain and range. It also discusses how to determine if two functions are inverses using their graphs and the horizontal line test.
The document discusses composition of functions. It defines composition of functions as using the output of one function as the input of another. It provides an example of composing two functions f and g, showing the steps of evaluating f(g(x)) and g(f(x)) at different values of x. Another example is given with two functions defined by sets of ordered pairs, finding the compositions fโg and gโf by evaluating them at different inputs and stating their domains and ranges.
The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, and division. Definitions of each operation are provided, along with examples of applying the operations to specific functions. Addition of functions involves adding the outputs of each function, subtraction involves subtracting the outputs, multiplication involves multiplying the outputs, and division involves dividing the outputs given the denominator function is not equal to 0. Several examples are worked through applying the different operations to functions like f(x)=2x and g(x)=-x+5. The examples also demonstrate evaluating composite functions and restricting domains as needed.
The document discusses determining the number and type of roots of polynomial equations. It states that every polynomial with degree greater than zero has at least one root in the set of complex numbers according to the Fundamental Theorem of Algebra. Descartes' Rule of Signs is introduced, which relates the number of changes in sign of a polynomial's terms to its possible positive and negative real roots. An example problem is worked through applying these concepts to determine the possible number of positive, negative, and imaginary roots.
This document discusses synthetic division and the remainder and factor theorems. It provides examples of using synthetic division to evaluate functions, determine the number of terms in a sequence, and factor polynomials. The key steps of synthetic division are shown, along with checking the remainder and determining common factors. Three examples are worked through to demonstrate these concepts.
This document discusses solving polynomial equations by factoring polynomials. It begins with essential questions and vocabulary about factoring polynomials and solving polynomial equations by factoring. It then provides the number of terms in a polynomial and the corresponding factoring technique that can be used. Examples of factoring various polynomials are also provided. The document aims to teach students how to factor polynomials and solve polynomial equations by factoring.
The document defines key terms and theorems related to trapezoids and kites. It provides definitions for trapezoid, bases, legs of a trapezoid, base angles, isosceles trapezoid, midsegment of a trapezoid, and kite. It also lists theorems about properties of isosceles trapezoids and kites. Two examples problems are included, one finding measures of an isosceles trapezoid and another showing a quadrilateral is a trapezoid.
The document discusses rhombi and squares. It defines a rhombus as a parallelogram with four congruent sides and gives its properties. A square is defined as a parallelogram with four right angles and four congruent sides. The document provides theorems for identifying rhombi and squares. It then gives examples of using the properties and theorems to determine if a shape is a rhombus, rectangle, or square.
The document discusses properties of rectangles. A rectangle is defined as a parallelogram with four right angles. The key properties are that opposite sides are parallel and congruent, opposite angles are congruent, and consecutive angles are supplementary. The diagonals of a rectangle bisect each other and are congruent. Theorems are presented regarding the diagonals of rectangles. Examples apply the properties of rectangles to find missing side lengths, angles, and diagonals. One example uses the distance formula and slope to determine if a quadrilateral is a rectangle.
The document discusses properties of parallelograms and provides examples of determining if a quadrilateral is a parallelogram. It defines four theorems for identifying parallelograms based on opposite sides, opposite angles, bisecting diagonals, and parallel/congruent sides. Examples solve systems of equations to find values of variables such that the quadrilaterals satisfy parallelogram properties. One example uses slopes of side segments to show a quadrilateral is a parallelogram due to parallel opposite sides.
The document discusses properties of parallelograms. It defines a parallelogram as a quadrilateral with two pairs of parallel sides. It then lists several properties of parallelograms: opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and if one angle is a right angle all angles are right angles. It also discusses properties of diagonals in parallelograms, including that diagonals bisect each other and divide the parallelogram into two congruent triangles. Several examples demonstrate using these properties to solve problems about parallelograms.
The document summarizes key concepts about polygons, including:
- The sum of the interior angles of a polygon with n sides is (n-2)180 degrees.
- The sum of the exterior angles of a polygon is 360 degrees.
- Examples are provided to demonstrate calculating sums of interior/exterior angles and finding missing angle measures using angle sums.
- Regular polygons are defined by their number of sides.
The document discusses analyzing graphs of polynomial functions. It provides examples of locating real zeros of polynomials using the location principle and estimating relative maxima and minima. Example 1 analyzes the polynomial f(x) = x^4 - x^3 - 4x^2 + 1 and locates its real zeros between consecutive integer values. Example 2 graphs the polynomial f(x) = x^3 - 3x^2 + 5 and estimates the x-coordinates of relative maxima and minima.
This document discusses polynomial functions. It defines key terms like polynomial in one variable, leading coefficient, and polynomial function. It provides examples of power functions of varying degrees like quadratic, cubic, quartic and quintic functions. The document also includes examples of evaluating polynomial functions, finding degrees and leading coefficients, graphing polynomial functions from tables of values, and describing properties of graphs.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
ย
(๐๐๐ ๐๐๐) (๐๐๐ฌ๐ฌ๐จ๐ง ๐)-๐๐ซ๐๐ฅ๐ข๐ฆ๐ฌ
๐๐ข๐ฌ๐๐ฎ๐ฌ๐ฌ ๐ญ๐ก๐ ๐๐๐ ๐๐ฎ๐ซ๐ซ๐ข๐๐ฎ๐ฅ๐ฎ๐ฆ ๐ข๐ง ๐ญ๐ก๐ ๐๐ก๐ข๐ฅ๐ข๐ฉ๐ฉ๐ข๐ง๐๐ฌ:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
๐๐ฑ๐ฉ๐ฅ๐๐ข๐ง ๐ญ๐ก๐ ๐๐๐ญ๐ฎ๐ซ๐ ๐๐ง๐ ๐๐๐จ๐ฉ๐ ๐จ๐ ๐๐ง ๐๐ง๐ญ๐ซ๐๐ฉ๐ซ๐๐ง๐๐ฎ๐ซ:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
ย
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the bodyโs response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
ย
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
This presentation was provided by Racquel Jemison, Ph.D., Christina MacLaughlin, Ph.D., and Paulomi Majumder. Ph.D., all of the American Chemical Society, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
2. ESSENTIAL QUESTION
โข How do you add, subtract, multiply, and divide to simplify
variable expressions?
โข Where youโll see this:
โข Sports, ๏ฌnance, photography, fashion, population
4. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
5. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
G
E
M
D
A
S
6. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
Grouping symbols
E
M
D
A
S
7. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
Grouping symbols
Exponents
M
D
A
S
8. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
Grouping symbols
Exponents
Multiplication
D
A
S
9. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
Grouping symbols
Exponents
Multiplication
Division
A
S
10. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
Grouping symbols
Exponents
Multiplication
Division } from left to right
A
S
11. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
Grouping symbols
Exponents
Multiplication
Division } from left to right
A ddition
S
12. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
Grouping symbols
Exponents
Multiplication
Division } from left to right
A ddition
Subtraction
13. VOCABULARY
1. Order of Operations: Allows for us to solve problems
to consistently achieve the same answers
Grouping symbols
Exponents
Multiplication
Division } from left to right
A ddition
Subtraction } from left to right
14. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
15. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
16. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
17. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
18. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
19. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
20. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x
7x +15
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
21. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x
7x +15
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
22. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x
7x +15
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
23. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x โ5x
7x +15
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
24. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x โ5x
7x +15
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
25. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x โ5x +5
7x +15
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
26. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x โ5x +5
7x +15 โ3x + 5
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
27. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x โ5x +5
7x +15 โ3x + 5
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
28. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x โ5x +5
7x +15 โ3x + 5
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
4x + 4y โ 7x + 7y
29. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x โ5x +5
7x +15 โ3x + 5
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
4x + 4y โ 7x + 7y
โ3x +11y
30. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x โ5x +5
7x +15 โ3x + 5
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
4x + 4y โ 7x + 7y
โ3x +11y
31. EXAMPLE 1
Simplify.
a. 5(x + 3) + 2x b. 2x โ 5(x โ1)
5x +15 +2x 2x โ5x +5
7x +15 โ3x + 5
c. 4(x + y) โ 7(x โ y) d. 2(mn + m) โ 5(mn โ n)
4x + 4y โ 7x + 7y 2mn + 2m โ 5mn + 5n
โ3x +11y
33. EXAMPLE 2
The ticket prices at Matt Mitarnowskiโs Googolplex are
$8.00 for regular admission and $5.50 for students and
seniors. For Saturdayโs ๏ฌrst show, 350 tickets were sold.
Write and simplify a variable expression for the total
admission fees of the tickets for that show.
34. EXAMPLE 2
The ticket prices at Matt Mitarnowskiโs Googolplex are
$8.00 for regular admission and $5.50 for students and
seniors. For Saturdayโs ๏ฌrst show, 350 tickets were sold.
Write and simplify a variable expression for the total
admission fees of the tickets for that show.
How many regular admission tickets were sold?
35. EXAMPLE 2
The ticket prices at Matt Mitarnowskiโs Googolplex are
$8.00 for regular admission and $5.50 for students and
seniors. For Saturdayโs ๏ฌrst show, 350 tickets were sold.
Write and simplify a variable expression for the total
admission fees of the tickets for that show.
How many regular admission tickets were sold?
r = regular tickets sold
36. EXAMPLE 2
The ticket prices at Matt Mitarnowskiโs Googolplex are
$8.00 for regular admission and $5.50 for students and
seniors. For Saturdayโs ๏ฌrst show, 350 tickets were sold.
Write and simplify a variable expression for the total
admission fees of the tickets for that show.
How many regular admission tickets were sold?
r = regular tickets sold
How many student and senior tickets were sold?
37. EXAMPLE 2
The ticket prices at Matt Mitarnowskiโs Googolplex are
$8.00 for regular admission and $5.50 for students and
seniors. For Saturdayโs ๏ฌrst show, 350 tickets were sold.
Write and simplify a variable expression for the total
admission fees of the tickets for that show.
How many regular admission tickets were sold?
r = regular tickets sold
How many student and senior tickets were sold?
350 โ r
39. So what was the total?
8.00r + 5.50(350 โ r )
40. So what was the total?
8.00r + 5.50(350 โ r )
8.00r +1925 โ 5.50r
41. So what was the total?
8.00r + 5.50(350 โ r )
8.00r +1925 โ 5.50r
2.50r +1925
42. So what was the total?
8.00r + 5.50(350 โ r )
8.00r +1925 โ 5.50r
2.50r +1925
The total admission fees were 2.50r + 1925 dollars
43. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
a. Two consecutive pages have a sum of 175. What
are the pages?
44. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
45. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
46. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175
47. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175
โ1
48. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175
โ1 โ1
49. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175
โ1 โ1
2n =174
50. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175
โ1 โ1
2n =174
2 2
51. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175
โ1 โ1
2n =174
2 2
n = 87
52. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175 87 is the ๏ฌrst page, 88 is the next.
โ1 โ1
2n =174
2 2
n = 87
53. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175 87 is the ๏ฌrst page, 88 is the next.
โ1 โ1
2n =174 Check:
2 2
n = 87
54. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175 87 is the ๏ฌrst page, 88 is the next.
โ1 โ1
2n =174 Check: 87+88=
2 2
n = 87
55. EXAMPLE 3
If a page in a book is numbered n, what is the
number of the next page?
n+1
a. Two consecutive pages have a sum of 175. What
are the pages?
n + (n +1) =175
2n +1=175 87 is the ๏ฌrst page, 88 is the next.
โ1 โ1
2n =174 Check: 87+88=175
2 2
n = 87
57. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
58. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
59. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
โ3
60. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
โ3 โ3
61. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
โ3 โ3
3n = 765
62. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
โ3 โ3
3n = 765
3 3
63. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
โ3 โ3
3n = 765
3 3
n = 255
64. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
โ3 โ3
3n = 765
3 3
n = 255
The pages are 255, 256, and 257.
65. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
โ3 โ3
3n = 765
3 3
n = 255
The pages are 255, 256, and 257.
Check:
66. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
โ3 โ3
3n = 765
3 3
n = 255
The pages are 255, 256, and 257.
Check: 255+256+257=
67. EXAMPLE 3
b. Three consecutive pages have a sum of 768.
n + (n +1) + (n + 2) = 768
3n + 3 = 768
โ3 โ3
3n = 765
3 3
n = 255
The pages are 255, 256, and 257.
Check: 255+256+257= 768
69. EXAMPLE 4
Find the area of the shaded region.
3(x โ 4)
3 5
6(x + 5)
Shaded area = Larger area - smaller area
70. EXAMPLE 4
Find the area of the shaded region.
3(x โ 4)
3 5
6(x + 5)
Shaded area = Larger area - smaller area
A = 5 โก6(x + 5)โค โ 3 โก3(x โ 4)โค
โฃ โฆ โฃ โฆ
71. EXAMPLE 4
Find the area of the shaded region.
3(x โ 4)
3 5
6(x + 5)
Shaded area = Larger area - smaller area
A = 5 โก6(x + 5)โค โ 3 โก3(x โ 4)โค
โฃ โฆ โฃ โฆ
A = 5 โก6x + 30 โค โ 3 โก3x โ12 โค
โฃ โฆ โฃ โฆ
72. EXAMPLE 4
Find the area of the shaded region.
3(x โ 4)
3 5
6(x + 5)
Shaded area = Larger area - smaller area
A = 5 โก6(x + 5)โค โ 3 โก3(x โ 4)โค
โฃ โฆ โฃ โฆ
A = 5 โก6x + 30 โค โ 3 โก3x โ12 โค
โฃ โฆ โฃ โฆ
A = 30x +150 โ 9x + 36
73. EXAMPLE 4
Find the area of the shaded region.
3(x โ 4)
3 5
6(x + 5)
Shaded area = Larger area - smaller area
A = 5 โก6(x + 5)โค โ 3 โก3(x โ 4)โค
โฃ โฆ โฃ โฆ
A = 5 โก6x + 30 โค โ 3 โก3x โ12 โค
โฃ โฆ โฃ โฆ
A = 30x +150 โ 9x + 36
A = 21x +186
74. EXAMPLE 4
Find the area of the shaded region.
3(x โ 4)
3 5
6(x + 5)
Shaded area = Larger area - smaller area
A = 5 โก6(x + 5)โค โ 3 โก3(x โ 4)โค
โฃ โฆ โฃ โฆ
A = 5 โก6x + 30 โค โ 3 โก3x โ12 โค
โฃ โฆ โฃ โฆ
A = 30x +150 โ 9x + 36
A = 21x +186 units2