Absolute Value Equations and Inequalities


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1 . Absolute Value Equations and Inequalities. OBJECTIVES 1. Solve an absolute value equation in one variable. Solve an absolute value inequality in one variable NOTE Technically we mean the distance between the point corresponding to x and the point corresponding to 0, the origin. Equations and inequalities may involve the absolute value notation in their statements. In this section we build on the tools developed in Sections.1 and.4 and on our earlier work with absolute value for the necessary solution techniques. Recall from Section 1. that the absolute value of x, written x, is the distance between x and 0 on the number line. Consider, for example, the absolute value equation x 4 This means that the distance between x and 0 is 4, as is pictured below. 4 units 4 units x 4 As the sketch illustrates, x 4 and x 4 are the two solutions for the equation. This observation suggests the more general statement. CAUTION p must be positive because an equation such as x has no solution. The absolute value of a quantity must always be equal to a nonnegative number. Rules and Properties: Absolute Value Equations Property 1 For any positive number p, if x p then x p or x p This property allows us to translate an equation involving absolute value to two linear equations that we can then solve separately. The following example illustrates. Example 1 Solving an Absolute Value Equation NOTE Add. NOTE Divide by. Solve for x: x 4 From Property 1 we know that x 4 is equivalent to the equations x 4 or x 4 x 6 x x x 91
2 9 CHAPTER LINEAR EQUATIONS AND INEQUALITIES CAUTION Be Careful! A common mistake is to solve only the equation x 4. You must solve both of the equivalent equations to find the two required solutions. The solutions are and. These solutions are easily checked by replacing x with and in the original absolute value equation. CHECK YOURSELF 1 Solve for x. 4x 1 9 An equation involving absolute value may have to be rewritten before you can apply Property 1. Consider the following example. Example Solving an Absolute Value Equation Solve for x: x 10 To use Property 1, we must first isolate the absolute value on the left side of the equation. This is easily done by subtracting from both sides for the result: x We can now proceed as before by using Property 1. NOTE Subtract. x or x x x 7 NOTE Divide by. x 1 x 7 The solution set is 1, 7. CHECK YOURSELF Solve for x. x 4 7 In some applications more than one absolute value is involved in an equation. Consider an equation of the form x y Because the absolute values of x and y are equal, x and y are the same distance from 0. This means they are either equal or opposite in sign. This leads to a second general property of absolute value equations.
3 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES SECTION. 9 Rules and Properties: Absolute Value Equations Property If x y then x y or x y Let s look at an application of this second property in our next example. Example Solving an Absolute Value Equation Solve for x: x 4 x By Property, we can write x 4 x or x 4 (x ) x 4 x x x 6 x x x 6 4x x x 1 The solution set is 1,. CHECK YOURSELF Solve for x. 4x 1 x We started this section by noting that the solution set for the equation x 4 consists of those numbers whose distance from the origin is equal to 4. Similarly, the solution set for the absolute value inequality x 4 consists of those numbers whose distance from the origin is less than 4, that is, all numbers between 4 and 4. The solution set is pictured below x 4
4 94 CHAPTER LINEAR EQUATIONS AND INEQUALITIES NOTE The solution set would be x 4 x 4 The solution set can be described by the compound inequality 4 x 4 and this suggests the following general statement. Rules and Properties: Absolute Value Inequalities Property 1 For any positive number p, if x p then p x p Let s look at an application of Property 1 in solving an absolute value inequality. Example 4 Solving an Absolute Value Inequality NOTE With Property 1 we can translate an absolute value inequality to an inequality not involving absolute value that can be solved by our earlier methods. Solve and graph the solution set of x From Property 1, we know that the given absolute value inequality is equivalent to the compound inequality x Solving as before, we isolate the variable in the center term. x 8 Add to all three parts. 1 x 4 Divide by. The solution set is x 1 x 4 The graph is shown below. NOTE Notice that the solution is an open interval on the number line CHECK YOURSELF 4 Solve and graph the solution set. x 4 8 We know that the solution set for the absolute value inequality x 4 consists of those numbers whose distance from the origin is less than 4. Now consider the solution set for x 4
5 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES SECTION. 9 It must consist of those numbers whose distance from the origin is greater than 4. The solution set is pictured below x 4 The solution set can be described by the compound inequality x 4 or x 4 and this suggests the following general statement. Rules and Properties: Absolute Value Inequalities Property For any positive number p, if x p then x p or x p Let s apply Property to the solution of an absolute value inequality. NOTE Again we translate the absolute value inequality to the compound inequality not involving absolute value. Example Solving an Absolute Value Inequality Solve and graph the solution set of x 8 From Property, we know that the given absolute value inequality is equivalent to the compound inequality x 8 or x 8 NOTE Add. NOTE Divide by. NOTE You could describe the solution set as Solving as before, we have x 6 or x 10 6 x x The solution set is x x 6 or x and the graph is shown below. x x 6 x x CHECK YOURSELF Solve and graph the solution set. x 9
6 96 CHAPTER LINEAR EQUATIONS AND INEQUALITIES The following chart summarizes our discussion of absolute value inequalities. NOTE As before, p must be a positive number. Type of Equivalent Inequality Inequality Graph of Solution Set ax b p p ax b p ax b p ax b p or ax b p r r s s CHECK YOURSELF ANSWERS 1.., 8., 4, 4 4. x 4 x x x or x 6 0 6
7 Name. Exercises Section Date Solve each of the following absolute value equations. 1. x. x 7 ANSWERS 1.. x 4. x 6... x x 0 7. x 7 8. x x x x x x 7 x x 8 x x x 17. Rewrite each of the following absolute value equations, and then solve the equations. 19. x 0. x x. x. x x
8 ANSWERS x 1 x x x x 4 0. x Solve each of the following absolute value equations. 1. x 1 x. x 1 x 4... x x x x x x 1 6. x x 7. x x 8. x 1 x x x 40. x 4 4 x Find and graph the solution set for each of the following absolute value inequalities. 41. x 4. x x x 4 4. x x 47. x x 7 98
9 ANSWERS 49. x 0. x 1. x 7 0. x 0. x 4. x x 4 6. x x x x x x x x 7 x On some popular calculators there is a special absolute value function key. It is usually labeled abs. To register an absolute value, you press this key and then put the desired expression in parentheses. For the expression x, enter abs(x ). Rewrite each expression in calculator form x 66. x x 68. x x x x 1 7. x
10 Answers 1.,. 1, , 10 9., , 1. 4, No solution 16, , 1. 0, 4.,. 0, , 7 9. No solution ,, All real numbers 41. x x 4. x x 7 or x x x or x x 10 x x x or x 8 1. No solution x 1 x 4. x x or x x x 4 or x x x x x 4 or x 6. 1 x 8 x abs(x ) 67. abs( * x ) 69. abs( * x ) * abs( * x 1) 100
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