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College Algebra
Tutorial 21:
Absolute Value Equations

 Learning Objectives

 After completing this tutorial, you should be able to:Solve absolute value equations.

 Introduction

 In this tutorial I will be stepping you through how to solve equationsthat have absolute values in them.  We will first go over the definitionof absolute value and then use that to help us solve our absolute valueequations.  You will find that when you have an absolute value expressionset equal to a positive number you will end up with two equations thatyou will need to solve to get your solutions.  Since the absolutevalue of any number other than zero is positive, it is not permissibleto set an absolute value expression equal to a negative number.  So,if your absolute value expression is set equal to a negative number, thenyou will have no solution.  The two main things that you need to knowfrom your past that will help you work the types of problems in this tutorialis how to solve linear and quadratic equations.  If you need a reviewon solving linear equations, feel free to .  If you need areview on solving quadratic equations, feel free to .  Ready, set, GO!!!

 Tutorial

 Absolute Value

 A lot of people know that when you take the absolute value of a numberthe answer is positive, but do you know why?  Lets find out:The absolute value measures the DISTANCE a number is away from theorigin (zero) on the number line.  No matter if the number isto the left (negative) or right (positive) of zero on the number line,the DISTANCE it is away from zero is going to be positive. Hence, the absolutevalue is always positive (or zero if you are taking the absolute valueof 0). Example1: What two numbers have an absolute value of 7?

 If you said 7 and -7, you are correct - good for you. Now I want to explain the thought behind it because this is going tohelp us to understand how to  solve absolute value equations. I really want to emphasize the fact that there are two numbers that arethe same distance away from the origin, the positive number and its opposite. The thought behind this is there are two places on the number line thatare 7 units away from zero - both 7 and -7.

 Solving an Absolute Value Equation

 Step 1:  Use thedefinition of absolute value to set up the equation without absolute values.

 If d is POSITIVE and |x|= d, thenx = d    OR    x =-d(two equations are set up) If d is NEGATIVE and |x|= d, thenNo solution This is because distance (d) can not be negative.

 Step 2: Solve the equationsset up in step 1.

 The equations in this tutorial will lead to either a linear or a quadraticequation.If you need a review on solving linear equations feel free to . If you need a review on solving quadratic equations feel free to .

 Example2: Solve the absolute value equation .

 We need to think about what value(s) are 7 units away from zeroon the number line.Using the definition of absolute value, there will be two equationsthat we will need to set up to get rid of the absolute value because thereare two places on the number line that are 7 units away from zero: 7 and-7.

 First equation: *Setting inside linear expression = to 7*Inv. of add. 5 is sub. 5 *Inv. of mult. by -3 is div. by -3

 Second equation: *Setting inside linear expression = to -7*Inv. of add. 5 is sub. 5 *Inv. of mult. by -3 is div. by -3

 When we plug -2/3 in for x, we end up withthe absolute value of 7 which is 7.  When we plug 4 in for x,we end up with the absolute value of -7 which is also 7.If you would have only come up with an answer of x= -2/3, you would not have gotten all solutions to this problem.Again it is important to note that we are using the definition of absolutevalue to set the two equations up.  Once you apply the definitionand set it up without the absolute value you just solve the linear equationas shown in Tutorial 14: LinearEquations in One Variable. There are two solutions to this absolute value equation:  -2/3and 4.

 Example3: Solve the absolute value equation .

 We need to think about what value(s) are 3 units away from zeroon the number line.Using the definition of absolute value, there will be two equationsthat we will need to set up to get rid of the absolute value because thereare two places on the number line that are 3 units away from zero: 3 and-3.

 First equation: *Setting inside quadratic expression = to3 *Quad. eq. in standard form*Factorthe trinomial*Use Zero-ProductPrinciple*Set 1st factor = 0 and solve      *Set 2nd factor = 0 and solve

 Second equation: *Setting inside quadratic expression = to-3 *Quad. eq. in standard form*Factorthe trinomial*Use Zero-ProductPrinciple*Set 1st factor = 0 and solve      *Set 2nd factor = 0 and solve

 When we plug -4 and 3 in for x, we endup with the absolute value of 3 which is 3.  When we plug -3 and 2in for x, we end up with the absolute valueof -3 which is also 3.If you would have only come up with an answer of x= -4 and 3, you would not have gotten all solutions to this problem.Again it is important to note that we are using the definition of absolutevalue to set the two equations up.  Once you apply the definitionand set it up without the absolute value you just solve the quadratic equationas shown in Tutorial 17:  QuadraticEquations. There is are four solutions to this absolute value equation: -4,3, -3, and 2.

 Example4: Solve the absolute value equation .

 Be careful on this one.  It is very tempting to set this up thesame way we did example 2 or 3 above, with two solutions.  However,note that the absolute value is set equal to a negative number. There isno value of x that we can plug in that willbe a solution because when we take the absolute value of the left sideit will always be positive or zero, NEVER negative.Answer: No solution.

 Example5: Solve the absolute value equation .

 We need to think about what value(s) are 0 units away from zeroon the number line.Using the definition of absolute value, there will be only one equationthat we will need to set up to get rid of the absolute value, because thereis only one place on the number line that is 0 units away from zero: 0.

 First equation: *Setting inside linear expression = to 0*Inv. of sub. 4 is add 4 *Inv. of mult. by 2 is div. by 2

 When we plug 2 in for x, we end up withthe absolute value of 0 which is 0. Again it is important to note that we are using the definition of absolutevalue to set the equation up.  Once you apply the definition and setit up without the absolute value you just solve the linear equation asshown in Tutorial 14: Linear Equationsin One Variable. There is only one solution to this absolute value equation: 2.

 Practice Problems

 These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of thesetypes of problems. Math works just like anythingelse, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots ofpractice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice.To get the most out of these, you should work the problem out onyour own and then check your answer by clicking on the link for the answer/discussionfor that  problem.  At the link you will find the answeras well as any steps that went into finding that answer.

 PracticeProblems 1a - 1c:Solve each absolute value equation.

 Need Extra Help on These Topics?

The following is a webpage that can assistyou in the topics that were covered on this page:

 Only look at problem 1.  Problem 1 at this webpage helpsyou with solving absolute value equations.

for somemore suggestions.

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July 24, 2002