1.7 solving absolute value equations part 1
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Absolute value equations can be solved by isolating the absolute value expression, setting up two equations where one drops the absolute value bars and the other negates the expression inside, and solving each equation. For example, to solve 6|5x + 2| = 312, one would isolate the absolute value term to get |5x + 2| = 52, set up the equations 5x + 2 = 52 and 5x + 2 = -52, solve to get x = 10 and x = -10.8, and check that the solutions satisfy the original equation.
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1.7 solving absolute value equations part 2
1. Absolute value equations can be solved by isolating the absolute value expression and setting up two separate equations, one with the expression unchanged and one with the expression negated.
2. The document provides examples of solving various absolute value equations including |x + 8| = 3, |y + 4| - 3 = 0, and 6|5x + 2| = 312.
3. Key steps are to isolate the absolute value expression, set up the two equations, solve each equation separately, and check the solutions.
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1. The document contains solutions to 20 math problems. It provides step-by-step working and explanations for finding values, solving equations, calculating probabilities and averages.
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(8) Lesson 2.2 - Solve Two-Step Equations
Here is a two-step equation example with steps to solve it:
2x + 3 = 11
Subtract 3 from both sides using the subtraction property of equality:
2x + 3 - 3 = 11 - 3
2x = 8
Divide both sides by 2 using the division property of equality:
2x/2 = 8/2
x = 4
To solve a two-step equation:
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2) Use multiplication or division to solve for the variable.
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2. The document provides examples of solving various absolute value equations including |x + 8| = 3, |y + 4| - 3 = 0, and 6|5x + 2| = 312.
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(8) Lesson 2.2 - Solve Two-Step Equations
Here is a two-step equation example with steps to solve it:
2x + 3 = 11
Subtract 3 from both sides using the subtraction property of equality:
2x + 3 - 3 = 11 - 3
2x = 8
Divide both sides by 2 using the division property of equality:
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Here are the steps to solve this system of equations using elimination:
1) Line up the equations:
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2) Multiply the top equation by -3:
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3 < x < 9
3 < x < 9
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こんなGemつかってますよ話
弊社社内チームミーティングでの資料です．ここ最近のRails開発（3.2系）で使っているGemの紹介など．
http://techracho.bpsinc.jp/author/morimorihoge にて他の記事も書いてますので，興味があればご覧下さい．
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The document discusses solving absolute value equations. It begins with examples of absolute value equations and their solutions. It then provides additional examples and special cases of absolute value equations, explaining how to set them up as cases and solve for the solutions. It also includes a word problem example about gas mileage and the absolute value equation used to find the minimum and maximum values.
Quadratic Equations in One Variables.pptx
The document discusses solving quadratic equations in one variable of the form ax^2 + bx + c = 0. It provides examples of quadratic equations and shows how to rewrite them in standard form. It then covers methods for solving quadratic equations, including extracting square roots, factoring, completing the square, and using the quadratic formula. It also discusses the nature of the roots based on the discriminant and provides rules for determining the sum and product of the roots.
6 4 Absolute Value And Graphing
This document reviews absolute value concepts including:
- Graphing absolute value equations and finding their domains and ranges
- Procedures for solving absolute value equations like |x-2|=7 and |x-10|=4.5
- Graphing the solutions to inequality equations like |x - (-5)| ≤ 8 and |x - 6| > 5
1.3 solving equations t
The document provides examples and explanations for solving different types of equations, including:
1) Polynomial equations through factoring or the quadratic formula.
2) Rational equations by clearing denominators.
3) Radical equations by squaring both sides to remove radicals.
4) Absolute value equations by recognizing that |x-c|=r implies x=c±r.
The document also discusses solving power equations, finding zeros and domains of functions, and using properties of absolute values.
Solving Quadratic-Equation.pptx
The document provides information about an online math class, including:
- A prayer asking God for guidance and wisdom as students wait to be taught.
- Reminders for online class such as turning on cameras, being on time, muting/unmuting audio.
- The weekly task of answering pretest questions from the module.
- An overview of the math topic for the first week of quarter 1 - quadratic equations. It discusses methods for solving quadratic equations such as extracting square roots, factoring, completing the square, and using the quadratic formula.
Harkeerit&Kyra
An absolute value of a number is its distance from zero on the number line. To write an absolute value, use bars (e.g. |5| = 5). To solve an absolute value equation, separate it into two cases - one where the expression inside the bars is positive and one where it is negative. Then solve each case separately and check the solutions. Basic graphs of absolute value functions form a V-shape, with restrictions so the y-values are always greater than or equal to 0. Shifting or stretching transformations can be applied.
Sept. 20
The document discusses absolute value, noting that absolute value represents the distance from zero regardless of direction. It provides examples of simplifying expressions involving absolute value, and explains that the absolute value of a variable depends on whether the variable is positive or negative.
4) Meaning of an Equation.ppt
This document discusses equations and how to solve them. It explains that an equation links two algebraic expressions and shows examples of equations involving variables and operations. It then demonstrates how to solve equations by reversing the operations, such as subtracting one side and dividing both sides. The goal is to learn how to understand equations and find the value of a variable by manipulating the operations.
Algebra 2 01-Systems of Linear Equations and Matrices (RW 2022).pptx
This document provides an overview of solving systems of linear equations using matrices. It introduces matrices and defines key matrix operations like addition, subtraction, and multiplication. It explains how to find the inverse of a matrix and use inverse matrices to solve systems of linear equations. Determinants are also introduced as a way to solve systems using Cramer's Rule or find the area of triangles. The document aims to teach students how to represent and manipulate matrices to solve real-world problems.
Core 3 Modulus 2
The document discusses solving simple modulus equations by setting the expressions equal and solving, either by isolating the variable or squaring both sides. It provides examples of solving equations of the form |ax + b| = c and |ax + b| = |cx + d|. The examples show setting the expressions equal, isolating the variable, and finding the solutions, or squaring both sides to remove the modulus and then solving the resulting equation.
Solving Absolute Value Equations and Inequalities.ppt
This document provides information and examples for solving absolute value equations and inequalities. It begins with definitions of absolute value and discusses how absolute value equations can have two solutions since the expression inside the absolute value can be positive or negative. Examples are provided for solving absolute value equations by setting the expression equal to both its positive and negative values. The document also discusses how to solve absolute value inequalities by splitting them into "and" or "or" statements and provides examples of solving and graphing various absolute value inequalities.
Solving Quadratic Equations
Provides example on how to solve quadratic equations using extracting square roots, factoring, completing the square and quadratic formula.
Solving equations
This document discusses how to solve equations and inequalities. It covers topics such as: balancing equations, combining like terms, expanding brackets, rearranging formulas to change the subject, representing inequalities on number lines, and solving one-step and two-step inequalities. Examples are provided to demonstrate each concept and practice problems are included for readers to try.
Final exam review #2
The document provides a review for an Algebra II final exam covering topics such as linear equations and inequalities, rational expressions, complex numbers, matrices, and polynomial functions. The review contains 50 practice problems across 7 sections testing different algebra concepts and skills needed to solve various types of problems.
Chapter-1-04032021-111422pm (2).pptx
The document discusses some preliminaries about algebraic equations and their solutions. It defines equations as statements of equality between two algebraic expressions. Variables can be involved, and numbers or values that satisfy the equation are called roots. The document also discusses identities, conditionals, and contradictions as types of equations. It provides examples of solving various types of first degree, quadratic, and absolute value equations.
Module 10 Topic 4 solving quadratic equations part 1
This document provides instruction on solving quadratic equations. It begins with an introduction explaining why quadratic equations are useful and includes a video example. It then defines quadratic equations and provides examples of solving quadratic equations by factoring and using the quadratic formula. The document uses examples and "Try It" problems for students to practice each step in solving quadratic equations.
GCSEYr9-SolvingQuadratics.pptx
Here are the steps to solve this equation by factorising:
1) Factorise the left hand side: 3(x - 1)(x + 2)
2) Set each factor equal to 0:
x - 1 = 0
x + 2 = 0
3) Solve for x:
x = 1
x = -2
Therefore, the solutions are x = 1 or x = -2.
Solving quadratic equations part 1
This document provides instruction on solving quadratic equations. It begins with an introduction explaining why quadratic equations are useful and includes a video example. It then defines quadratic equations and shows students how to identify the coefficients a, b, and c. The bulk of the document demonstrates two methods for solving quadratic equations: factoring and using the quadratic formula. It includes examples of each method and practice problems for students to work through. The goal is to teach students how to solve quadratic equations through factoring and using the formula.
Thursday, september 26, 2013
The document provides instructions and examples for solving integer equations and working with absolute values. It discusses key concepts like:
- The rules for adding, subtracting, multiplying and dividing integers
- Classifying numbers as natural, whole, integer, rational or real
- Understanding that the absolute value of a number represents its distance from zero
- Solving problems involving absolute values and performing operations inside and outside of absolute value signs
Absolute Value Equations
The document discusses solving absolute value equations. It explains that absolute value is the distance a number is from 0, and provides examples. It then states that when solving absolute value equations, two separate equations must be created to account for the number inside the absolute value being positive or negative. Steps are provided for solving sample absolute value equations.
Similar to 1.7 solving absolute value equations part 1 (20)
Algebra 2 01-Systems of Linear Equations and Matrices (RW 2022).pptx
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Module 10 Topic 4 solving quadratic equations part 1
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3.4 linear programming
The document discusses linear programming problems. It provides two examples of finding the minimum and maximum values of objective functions subject to certain constraints. In the first example, the objective is to minimize or maximize the function f(x,y)=3x-2y within the constraints 1≤x≤5, y≥2, and y≤x+3. In the second example, the objective is to minimize or maximize the function f(x,y)=4x+3y within the constraints y≥-x+2, y≤(1/4)x+2, and y≥2x-5. Both examples solve the problems by finding the vertices of the constrained regions and calculating the objective
3.3 a writing systems of linear inequalities
The document discusses how to graph and write systems of linear inequalities. It explains how to graph systems by determining whether lines should be dotted or solid based on the inequality symbols, and whether shading should be above or below lines based on 'y >' or 'y <'. It also explains how to write the system of inequalities given a graph by writing equations for each line and changing the equations to inequalities based on whether lines are dotted or solid and whether shading is above or below lines. Examples of these processes are provided.
3.3 a writing a systems of inequalities
This document discusses how to graph systems of linear inequalities by:
1. Writing each inequality in the system in slope-intercept form (y=mx+b)
2. Graphing the lines for each inequality, making dashed lines for > or < and solid for ≥ or ≤
3. Shading the correct region for each line (above for > and ≥, below for < and ≤)
4. The solution to the system is the overlapping region shaded by both inequalities.
3.2 a solving systems algebraically
Here are the steps to solve this system of linear equations using the elimination method:
1) Line up the equations:
-4x - 8y = 16
6x + 12y = -24
2) Add a multiple of one equation to the other to eliminate x:
-4x - 8y = 16
6x + 12y = -24
Add -1.5 times the top equation to the bottom:
-4x - 8y = 16
0 - 3y = -40
3) Solve for y:
-3y = -40
y = 13
4) Substitute y = 13 back into either original equation to find x:
-4x
3.1 b solving systems graphically
This document provides instructions for students to solve systems of linear equations by graphing. Students are asked to write equations in slope-intercept form when possible, use slope and y-intercept to determine the number of solutions (one, many, or no solutions), and name each system based on its number of solutions. The document assigns textbook problems #35-52 for practice and is dated for Friday, October 7th.
3.1 solving systems graphically
The document discusses solving systems of linear equations by graphing. It explains that a system of linear equations contains two or more linear equations using the same variables. To solve a linear system graphically, the two equations must be put into slope-intercept form and graphed on the same coordinate plane. The point where the lines intersect satisfies both equations and is the solution to the system. The solution must be verified by substituting the coordinates into the original equations.
2.8 absolute value functions
1. The document discusses absolute value functions of the form y = a|x - h| + k, where a ≠ 0.
2. It explains how to find the vertex (h, k) and line of symmetry x = h of an absolute value function graph.
3. It also describes how the graph opens up or down depending on whether a is positive or negative, and how the width compares to y = |x| depending on the value of |a|.
2.6 graphing linear inequalities
This document discusses graphing linear inequalities. It has two objectives: 1) determining if an ordered pair is a solution to an inequality, and 2) graphing linear inequalities in one and two variables.
2.5 correlation & best fitting lines
This document discusses identifying correlation in data by plotting it on a scatter plot and fitting a best-fit line. It explains that a positive slope indicates positive correlation, a negative slope indicates negative correlation, and no consistent slope indicates little correlation. It also introduces the correlation coefficient r as a measure of correlation strength, with values closer to 1 or -1 indicating stronger correlation.
2.4 writing linear equations
This document discusses writing and working with linear equations in slope-intercept form (y=mx+b). It provides examples of:
1) Writing linear equations from the slope (m) and y-intercept (b)
2) Finding the slope and y-intercept of given linear equations
3) Writing linear equations given two points or given a slope and point.
2.2 slope
1. The document discusses slope and rate of change, including how to calculate slope from two points on a line and how to interpret positive, negative, zero, and undefined slopes.
2. It provides examples of finding the slope of lines from graphs and points, and discusses how to identify parallel and perpendicular lines based on their slopes.
3. The key concepts are how to determine the steepness of a line from its slope, and that parallel lines have the same slope while perpendicular lines have slopes that are opposite reciprocals.
2.1 a relations and functions
1. The document discusses functions and their graphs.
2. It defines a vertical line test to determine if a graph represents a function and provides examples of graphs that both do and do not represent functions.
3. It introduces function notation using f(x) and provides examples of evaluating functions for given values of x.
2.7 a graphing piecewise functions
Graphing piecewise functions involves graphing step-functions. Step-functions are functions defined by different formulas on different intervals of the domain that create a graph that looks like steps. The objective is to graph these types of piecewise functions by determining the formulas used on each interval and plotting the corresponding points to create the step-like graph.
Interval notation
Interval notation is used to represent sets of numbers graphically or through inequalities. The objectives are to become familiar with interval notation and to use it to represent sets of numbers given in graphical or inequality form in 3 sentences or fewer.
1.6 solving inequalities
The document discusses how to solve simple and compound inequalities. It explains that solving inequalities follows the same process as solving equations, but requires paying attention to the inequality sign and graphing the solution set. It provides examples of solving different types of inequalities, such as those with addition, subtraction, multiplication, and division, noting that the inequality sign may need to be reversed when multiplying or dividing by a negative number.
1.5 problem solving using algebraic models
The document outlines a general problem solving plan for real-life problems involving four steps: 1) write a verbal model, 2) assign labels, 3) write an algebraic model, and 4) solve the algebraic model. It then provides two examples of using this plan to solve word problems about the average speed of a bullet train and the flow rate of a faucet.
1.4 rewriting equations and formulas
This document provides instructions for rewriting equations and formulas. It discusses rewriting equations with more than one variable by substituting values and then solving for the remaining variable. Common formulas are also restructured by solving for a specified variable through steps like multiplication, division, addition and subtraction. Worked examples demonstrate solving equations for variables like y, h, and b1.
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1.7 solving absolute value equations part 1
- 1. Solving Absolute Value Equations Today’s objective: 1. I will solve absolute value equations. 2. I will use absolute value equations to solve real-life problems.
- 2. Solving Absolute Value Equations • Absolute value is denoted by the bars |3|. • Absolute value represents the distance a number is from 0. Thus, it is always positive. • |8| = 8 and |-8| = 8
- 3. Solving absolute value equations • First, isolate the absolute value expression. • Set up two equations to solve. • For the first equation, drop the absolute value bars and solve the equation. • For the second equation, drop the bars, negate the opposite side, and solve the equation. • Always check the solutions.
- 4. 6|5x + 2| = 312 • Isolate the absolute value expression by dividing by 6. 6|5x + 2| = 312 |5x + 2| = 52 • Set up two equations to solve. 5x + 2 = 52 5x + 2 = -52 5x = 50 5x = -54 x = 10 or x = -10.8 •Check: 6|5x + 2| = 312 6|5x + 2| = 312 6|5(10)+2| = 312 6|5(-10.8)+ 2| = 312 6|52| = 312 6|-52| = 312 312 = 312 312 = 312
- 5. 3|x + 2| -7 = 14 • Isolate the absolute value expression by adding 7 and dividing by 3. 3|x + 2| -7 = 14 3|x + 2| = 21 |x + 2| = 7 • Set up two equations to solve. x+2=7 x + 2 = -7 x=5 or x = -9 •Check: 3|x + 2| - 7 = 14 3|x + 2| -7 = 14 3|5 + 2| - 7 = 14 3|-9+ 2| -7 = 14 3|7| - 7 = 14 3|-7| -7 = 14 21 - 7 = 14 21 - 7 = 14 14 = 14 14 = 14