1) Solve each inequality and graph the solution. There are 16 inequalities involving various variables such as z, d, v, m, f, w, b, t, u, z, b, k, a, b, k, j. The inequalities include addition, subtraction, multiplication and division operations.
1. The document provides 25 word problems involving inequalities that must be solved and graphed.
2. It then provides 4 additional practice problems involving solving and graphing inequalities.
3. The key information is extracting the mathematical expressions from the word problems, writing them as inequalities, solving for the variable, and graphing the solution on a number line.
Finding solution set of quadratic inequalitiesMartinGeraldine
The document shows the steps to solve the inequality 2x(x-1) ≤ 4 by writing it in standard form, factorizing, and solving for x. The solution set is obtained as the intervals [-1,2]. A similar process is shown for the inequality x^2 ≥ 4, with the solution set given as the intervals [2,-2].
This document contains 10 math problems involving simplifying repeating decimals to fractions and performing basic arithmetic operations. The problems include rewriting repeating decimals as fractions, adding fractions, dividing fractions, and multiplying a fraction by a whole number.
This document contains a quiz reviewing writing and graphing inequalities, solving linear inequalities, and factoring sums. There are 25 problems in total where students are asked to write inequalities, graph inequalities on number lines, solve linear inequalities algebraically and graphically, and factor sums as the greatest common factor times the sum.
Maths Olympiad - Try this prime-factor questionKathleen Ong
The document finds the number of distinct prime factors of the sum of 2, 22, 23, ..., 215, 216. It factors the sum into prime factors, showing the number of distinct prime factors is 5, since the prime factorization includes the distinct prime factors of 2, 3, 5, and 17.
This document presents the steps to solve a system of 3 equations with 3 unknowns (x, y, z) using Gaussian elimination. It starts with the original system of equations in matrix form. It then applies row operations such as multiplication, addition, and division of rows to put the matrix in reduced row echelon form. The final form reveals the solutions for x, y, and z. These solutions are verified by substituting them back into the original system of equations.
The document discusses the motivation for using B-trees to store large datasets that do not fit into main memory. It notes that while binary search trees provide logarithmic-time performance, disk access times are significantly slower than memory. B-trees are designed to group related data together to minimize disk I/O and improve performance. The document defines B-trees as m-way search trees where nodes can have up to m children, leaves are on the same level, and operations like insertion and deletion involve splitting or merging nodes to balance the tree.
This document describes the bottom-up mergesort algorithm with sharing. It shows an example of sorting the numbers 5, 2, 7, 4, 1, 8, 3 step-by-step. It then analyzes the amortized cost of adding new numbers and of the entire sorting process, showing both take O(log n) time and O(n) time respectively.
1. The document provides 25 word problems involving inequalities that must be solved and graphed.
2. It then provides 4 additional practice problems involving solving and graphing inequalities.
3. The key information is extracting the mathematical expressions from the word problems, writing them as inequalities, solving for the variable, and graphing the solution on a number line.
Finding solution set of quadratic inequalitiesMartinGeraldine
The document shows the steps to solve the inequality 2x(x-1) ≤ 4 by writing it in standard form, factorizing, and solving for x. The solution set is obtained as the intervals [-1,2]. A similar process is shown for the inequality x^2 ≥ 4, with the solution set given as the intervals [2,-2].
This document contains 10 math problems involving simplifying repeating decimals to fractions and performing basic arithmetic operations. The problems include rewriting repeating decimals as fractions, adding fractions, dividing fractions, and multiplying a fraction by a whole number.
This document contains a quiz reviewing writing and graphing inequalities, solving linear inequalities, and factoring sums. There are 25 problems in total where students are asked to write inequalities, graph inequalities on number lines, solve linear inequalities algebraically and graphically, and factor sums as the greatest common factor times the sum.
Maths Olympiad - Try this prime-factor questionKathleen Ong
The document finds the number of distinct prime factors of the sum of 2, 22, 23, ..., 215, 216. It factors the sum into prime factors, showing the number of distinct prime factors is 5, since the prime factorization includes the distinct prime factors of 2, 3, 5, and 17.
This document presents the steps to solve a system of 3 equations with 3 unknowns (x, y, z) using Gaussian elimination. It starts with the original system of equations in matrix form. It then applies row operations such as multiplication, addition, and division of rows to put the matrix in reduced row echelon form. The final form reveals the solutions for x, y, and z. These solutions are verified by substituting them back into the original system of equations.
The document discusses the motivation for using B-trees to store large datasets that do not fit into main memory. It notes that while binary search trees provide logarithmic-time performance, disk access times are significantly slower than memory. B-trees are designed to group related data together to minimize disk I/O and improve performance. The document defines B-trees as m-way search trees where nodes can have up to m children, leaves are on the same level, and operations like insertion and deletion involve splitting or merging nodes to balance the tree.
This document describes the bottom-up mergesort algorithm with sharing. It shows an example of sorting the numbers 5, 2, 7, 4, 1, 8, 3 step-by-step. It then analyzes the amortized cost of adding new numbers and of the entire sorting process, showing both take O(log n) time and O(n) time respectively.
The document is notes from a math lesson on multiplying integers. It includes examples of multiplying integers with the same sign and opposite signs. The lesson notes show that multiplying integers with the same sign results in a positive product, while multiplying integers with opposite signs results in a negative product. The document also contains homework answers and problems for understanding multiplication of integers.
1) The document discusses simplifying, multiplying, dividing, adding and subtracting rational expressions and radical functions. It provides examples and steps for simplifying, multiplying, dividing, rational expressions and finding common denominators when adding or subtracting them.
2) The document also discusses dividing polynomials using long division and provides examples. It explains how to add and subtract rational expressions with polynomial denominators by finding the least common denominator.
3) Additional examples are given for adding and subtracting rational expressions with binomial and polynomial denominators. Steps are outlined for finding the least common denominator in order to combine like terms in the numerator.
The document presents the multiplication table for 2, showing the sum of 10 twos equaling 20. It then lists the multiplication problems 1 times 2 through 10 times 2, leaving the answers blank to be filled in. This is repeated below to provide an example multiplication table to work through.
The document describes the formulation of an optimization problem to maximize the objective function f(x1, x2) = 10x1 + 20x2 subject to several constraints. It introduces additional variables x3, x4, x5, x6 to transform the constraints into equality constraints. The problem is then restated using only variables x1, x2, x5 by eliminating x3, x4, x6. Finally, a penalty term -Mx6 is added to the objective function to encourage x6 = 0 and further simplify the problem.
Review Sheet B Substitution And Solving Inequalitiesvmonacelli
The document contains a review sheet on substitution and solving inequalities with 10 problems. It begins with 4 problems simplifying expressions using substitution when given a value for the variable. It then has 2 problems solving one-step inequalities and 2 problems solving two-step inequalities. It concludes with a problem to graph an inequality.
1) <7 and all angles corresponding to <7 are <1, <2, <3, <4, <5, <6, <7, <8, <9, <10, <11, <12, <13, <14, <15, <16.
2) A matching pair of alternate interior angles is <5 and <8.
3) A matching pair of alternate exterior angles is <13 and <16.
4) A set of consecutive interior angles is <1, <2, <7, <8.
5) The measure of <1 is 60°. The measure of <2 is 110°.
This document discusses manipulations of rational expressions. It begins by differentiating between rational numbers and rational expressions. The objectives are to simplify, multiply, divide, add and subtract rational expressions. Examples are provided to distinguish rational numbers from expressions. Steps are outlined for simplifying expressions, including factorizing and cancelling common factors. Students work through practice problems in groups and individually. The document concludes by assigning further study on multiplying and dividing rational expressions.
This document provides a review for a quiz on solving inequalities. It contains 11 practice problems involving solving linear inequalities algebraically and graphing their solutions on number lines. The problems cover solving single-variable inequalities for x, compound inequalities, and choosing the graph that represents the solution set of a given inequality.
The document provides steps for dividing out common factors and factoring rational expressions:
1) Identify common factors in the numerator and denominator and divide them out.
2) Factor the numerator and denominator by identifying common factors in each expression.
3) Divide the factored numerator and denominator, simplifying the expression.
This is part of a Mathematics Quiz that we conducted at our school. For more information, visit http://sneeze10.blogspot.com/2012/11/mathematics-quiz-at-school.html
The daily lesson plan outlines a mathematics lesson for year 6 pupils on dividing fractions, with the learning objective being that pupils will learn to divide fractions with a whole number and a fraction. The lesson includes a paper-folding activity, interactive questions to check understanding, and a plenary game where pupils will roll dice to generate division problems and find quotients in groups. The vocabulary, previous knowledge, problem, and learning outcome are also defined.
The document provides examples of cubic numbers from 1 to 10, which are numbers that are the result of multiplying a single integer by itself three times. It lists the cubic numbers 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000 that correspond to cubing the integers from 1 to 10. It also includes a simple Python-like pseudocode to print the cubic numbers from 1 to 10.
The document describes the optimization of a linear function subject to linear constraints. It involves defining the objective function and constraints, substituting variables to eliminate parameters, and determining the optimal solution satisfies all constraints. The optimal solution is found to be x1=0, x2=5, x3=0, x4=0, x5=12.
This document shows a series of addition problems with the numbers 2, 1, and 3. It adds up combinations of those numbers on separate lines and shows the equals sign after each one is calculated to equal a total. The document does not provide any other context around the meaning or purpose.
This document contains math problems involving multiplication, division, factors, and multiples. It lists 6 multiplication problems, 6 division problems, and asks to find the factors of 35 and 24 and the first three common multiples of 3 and 6. It then introduces fractions as the next math concept covered.
This document provides an introduction to quadratic equations including:
- The definition of a quadratic equation as an equation of the form ax^2 + bx + c = 0 where a ≠ 0.
- Methods for solving quadratic equations including factorization, completing the square, and the quadratic formula.
- Examples of solving quadratic equations by each method and deriving the quadratic formula.
- Applications of quadratic equations such as finding the time it takes a ball to hit the ground after being thrown up and determining the width of metal needed to cut a frame with a given area.
The document presents a system of 3 equations with variables X, Y, and Z:
(X2)’ – 3Y + 5 Z/3 =1
-X + 4 Y/3 - Z = 0
3 x/2 + Y – Z = 0
It then solves for the values of the variables by manipulating and combining the equations. The solutions are:
X = 5/11
Y = 1/22
Z = 3/22
This document contains practice problems asking students to find the lateral area and surface area of cones and pyramids. There are 12 problems total broken into 4 sections - finding the lateral area of cones, surface area of cones in terms of pi, lateral area of pyramids, and surface area of pyramids. Students are asked to show their work and provide answers to the nearest specified value.
This math worksheet asks students to find the lateral and surface areas of various prisms and cylinders. For problems 1-6, students are asked to find the lateral area of cylinders to the nearest tenth. Problems 7-12 ask students to find both the lateral and surface areas of different prisms, rounding answers to the nearest whole number. The final problems 13-15 require finding the surface area of cylinders in terms of pi.
The document is notes from a math lesson on multiplying integers. It includes examples of multiplying integers with the same sign and opposite signs. The lesson notes show that multiplying integers with the same sign results in a positive product, while multiplying integers with opposite signs results in a negative product. The document also contains homework answers and problems for understanding multiplication of integers.
1) The document discusses simplifying, multiplying, dividing, adding and subtracting rational expressions and radical functions. It provides examples and steps for simplifying, multiplying, dividing, rational expressions and finding common denominators when adding or subtracting them.
2) The document also discusses dividing polynomials using long division and provides examples. It explains how to add and subtract rational expressions with polynomial denominators by finding the least common denominator.
3) Additional examples are given for adding and subtracting rational expressions with binomial and polynomial denominators. Steps are outlined for finding the least common denominator in order to combine like terms in the numerator.
The document presents the multiplication table for 2, showing the sum of 10 twos equaling 20. It then lists the multiplication problems 1 times 2 through 10 times 2, leaving the answers blank to be filled in. This is repeated below to provide an example multiplication table to work through.
The document describes the formulation of an optimization problem to maximize the objective function f(x1, x2) = 10x1 + 20x2 subject to several constraints. It introduces additional variables x3, x4, x5, x6 to transform the constraints into equality constraints. The problem is then restated using only variables x1, x2, x5 by eliminating x3, x4, x6. Finally, a penalty term -Mx6 is added to the objective function to encourage x6 = 0 and further simplify the problem.
Review Sheet B Substitution And Solving Inequalitiesvmonacelli
The document contains a review sheet on substitution and solving inequalities with 10 problems. It begins with 4 problems simplifying expressions using substitution when given a value for the variable. It then has 2 problems solving one-step inequalities and 2 problems solving two-step inequalities. It concludes with a problem to graph an inequality.
1) <7 and all angles corresponding to <7 are <1, <2, <3, <4, <5, <6, <7, <8, <9, <10, <11, <12, <13, <14, <15, <16.
2) A matching pair of alternate interior angles is <5 and <8.
3) A matching pair of alternate exterior angles is <13 and <16.
4) A set of consecutive interior angles is <1, <2, <7, <8.
5) The measure of <1 is 60°. The measure of <2 is 110°.
This document discusses manipulations of rational expressions. It begins by differentiating between rational numbers and rational expressions. The objectives are to simplify, multiply, divide, add and subtract rational expressions. Examples are provided to distinguish rational numbers from expressions. Steps are outlined for simplifying expressions, including factorizing and cancelling common factors. Students work through practice problems in groups and individually. The document concludes by assigning further study on multiplying and dividing rational expressions.
This document provides a review for a quiz on solving inequalities. It contains 11 practice problems involving solving linear inequalities algebraically and graphing their solutions on number lines. The problems cover solving single-variable inequalities for x, compound inequalities, and choosing the graph that represents the solution set of a given inequality.
The document provides steps for dividing out common factors and factoring rational expressions:
1) Identify common factors in the numerator and denominator and divide them out.
2) Factor the numerator and denominator by identifying common factors in each expression.
3) Divide the factored numerator and denominator, simplifying the expression.
This is part of a Mathematics Quiz that we conducted at our school. For more information, visit http://sneeze10.blogspot.com/2012/11/mathematics-quiz-at-school.html
The daily lesson plan outlines a mathematics lesson for year 6 pupils on dividing fractions, with the learning objective being that pupils will learn to divide fractions with a whole number and a fraction. The lesson includes a paper-folding activity, interactive questions to check understanding, and a plenary game where pupils will roll dice to generate division problems and find quotients in groups. The vocabulary, previous knowledge, problem, and learning outcome are also defined.
The document provides examples of cubic numbers from 1 to 10, which are numbers that are the result of multiplying a single integer by itself three times. It lists the cubic numbers 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000 that correspond to cubing the integers from 1 to 10. It also includes a simple Python-like pseudocode to print the cubic numbers from 1 to 10.
The document describes the optimization of a linear function subject to linear constraints. It involves defining the objective function and constraints, substituting variables to eliminate parameters, and determining the optimal solution satisfies all constraints. The optimal solution is found to be x1=0, x2=5, x3=0, x4=0, x5=12.
This document shows a series of addition problems with the numbers 2, 1, and 3. It adds up combinations of those numbers on separate lines and shows the equals sign after each one is calculated to equal a total. The document does not provide any other context around the meaning or purpose.
This document contains math problems involving multiplication, division, factors, and multiples. It lists 6 multiplication problems, 6 division problems, and asks to find the factors of 35 and 24 and the first three common multiples of 3 and 6. It then introduces fractions as the next math concept covered.
This document provides an introduction to quadratic equations including:
- The definition of a quadratic equation as an equation of the form ax^2 + bx + c = 0 where a ≠ 0.
- Methods for solving quadratic equations including factorization, completing the square, and the quadratic formula.
- Examples of solving quadratic equations by each method and deriving the quadratic formula.
- Applications of quadratic equations such as finding the time it takes a ball to hit the ground after being thrown up and determining the width of metal needed to cut a frame with a given area.
The document presents a system of 3 equations with variables X, Y, and Z:
(X2)’ – 3Y + 5 Z/3 =1
-X + 4 Y/3 - Z = 0
3 x/2 + Y – Z = 0
It then solves for the values of the variables by manipulating and combining the equations. The solutions are:
X = 5/11
Y = 1/22
Z = 3/22
This document contains practice problems asking students to find the lateral area and surface area of cones and pyramids. There are 12 problems total broken into 4 sections - finding the lateral area of cones, surface area of cones in terms of pi, lateral area of pyramids, and surface area of pyramids. Students are asked to show their work and provide answers to the nearest specified value.
This math worksheet asks students to find the lateral and surface areas of various prisms and cylinders. For problems 1-6, students are asked to find the lateral area of cylinders to the nearest tenth. Problems 7-12 ask students to find both the lateral and surface areas of different prisms, rounding answers to the nearest whole number. The final problems 13-15 require finding the surface area of cylinders in terms of pi.
This document contains 30 problems involving multiplication of exponential terms. Each problem contains terms with exponents that need to be combined using the properties of exponents, such as (ab)n = anbn and (a/b)n = an/bn. The goal is to simplify each exponential expression into a single term.
1. This document contains a practice worksheet with 72 problems involving operations with zero and negative exponents. The problems include simplifying expressions, evaluating expressions for given values of variables, writing numbers as powers of 10 with negative exponents, writing expressions as decimals, and evaluating expressions with multiple variables.
This document contains a practice worksheet with 72 math problems involving operations with zero and negative exponents, simplifying expressions, and evaluating expressions for given values of variables. The problems cover simplifying expressions, writing numbers as powers of 10 using negative exponents, converting expressions to decimals, and evaluating expressions when given specific values for variables.
7.6 systems of inequalities word problemsMsKendall
This document contains 10 multi-part math word problems involving systems of inequalities. The problems ask students to determine possible dimensions of a habitat given area constraints, possible scores in a basketball game given point constraints, possible work hour combinations at two jobs given earnings and time constraints, possible food item quantities given spending constraints, possible fundraising quantities given earnings constraints, and possible clothing item quantities given spending constraints. Students are asked to provide three possible solutions or combinations for each problem.
7.6 solving systems of inequalities hw (mixed forms)MsKendall
The document contains instructions to graph 9 systems of inequalities involving two variables, x and y. The systems include linear inequalities like y > 2x + 1 and nonlinear inequalities like 3x + 5y < 10. The goal is to sketch the regions defined by each system of inequalities on a standard x-y coordinate plane.
7.6 solving systems of inequalities cw (mixed forms)MsKendall
This document contains instructions for graphing systems of inequalities on a coordinate plane. It lists 8 systems of inequalities and provides blank coordinate planes for graphing the solution sets of each system. The systems involve various combinations of linear inequalities in the variables x and y. The task is to graph the region defined by satisfying all the inequalities in each system on the corresponding coordinate plane.
7.6 solving systems of inequalities hw (mixed forms)MsKendall
The document contains instructions to graph 9 systems of inequalities involving two variables, x and y. The systems include linear inequalities like y > 2x + 1 and nonlinear inequalities like 3x + 5y < 10. The goal is to sketch the regions defined by each system of inequalities on a standard x-y coordinate plane.
7.6 solving systems of inequalities cw (mixed forms)MsKendall
This document contains instructions for graphing systems of inequalities on a coordinate plane. It lists 8 systems of inequalities and provides blank coordinate planes for graphing the solution sets of each system. The systems involve various combinations of linear inequalities in the variables x and y. The task is to graph the region defined by satisfying all the inequalities in each system on the corresponding coordinate plane.
This document provides practice problems involving identifying nets that fold into three-dimensional shapes like cubes and pyramids. It includes choosing the correct nets to make cubes or pyramids with square bases, matching 3D shapes to their corresponding nets, and using Euler's formula to find missing numbers for shapes specified by their faces, edges and vertices. The practice problems are accompanied by diagrams of potential nets labeled with dimensions.
The document contains a practice problem involving systems of linear inequalities. There are 21 problems where the learner is asked to write systems of inequalities to model word problems, graph the systems to visualize the solution sets, and provide possible solutions. The document focuses on representing and solving word problems using systems of linear inequalities.
This document contains a worksheet with graphing systems of inequalities problems. The worksheet asks students to determine which points are solutions to an inequality graph, solve individual inequalities, write inequality sentences for graphs, and determine if points are in the solution regions of graphs.
The document contains practice problems involving graphing linear inequalities on a number line. There are 27 problems total, asking to graph linear inequalities like y ≥ -4, x + y < -2, and 6x - 4y > -16. It also contains 3 word problems for each of which the student is asked to write a linear inequality describing the situation, graph it, and provide two possible solutions.
This document contains a math practice worksheet with 22 problems asking students to find the area of various polygons using trigonometry. The polygons include equilateral triangles, squares, regular hexagons, regular pentagons, regular octagons, and regular decagons with given apothems or radii. Some problems also ask students to find the area of unspecified triangles.
This document contains a math practice worksheet with 22 problems calculating the areas of various polygons using trigonometry. The problems involve finding the areas of equilateral triangles, squares, regular hexagons, regular pentagons, regular octagons, and regular decagons given their apothems or radii. Other problems calculate the areas of triangles. The final problems calculate the areas of regular polygons in real world contexts like dog pens, swimming pools, and patios.
The document contains 27 word problems from various categories including systems of linear equations, tickets/admissions, coins, digits, break even, wind/current. The problems provide relevant context and numerical information to solve multi-step math word problems across different domains.
The document provides a review of the three methods to solve systems of equations: graphing, substitution, and elimination. It includes examples of systems of equations to solve using each method. Checkpoint questions are provided to have the student practice solving systems of equations by graphing, substitution, and elimination.