Section 3: Absolute Value Equations

# Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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• davidvictor
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Section 4.3 Absolute Value Equations 373 Version: Fall 2007 4.3 Absolute Value Equations In the previous section, we defined | x | = x, if x < 0 . x, if x 0 , (1) and we saw that the graph of the absolute value function defined by f ( x ) = | x | has the “V-shape” shown in Figure 1 . x 10 y 10 Figure 1. The graph of the absolute value function f ( x ) = | x | . It is important to note that the equation of the left-hand branch of the “V” is y = x . Typical points on this branch are ( 1 , 1) , ( 2 , 2) , ( 3 , 3) , etc. It is equally important to note that the right-hand branch of the “V” has equation y = x . Typical points on this branch are (1 , 1) , (2 , 2) , (3 , 3) , etc. Solving | x | = a We will now discuss the solutions of the equation | x | = a. There are three distinct cases to discuss, each of which depends upon the value and sign of the number a . Case I: a < 0 If a < 0 , then the graph of y = a is a horizontal line that lies strictly below the x -axis, as shown in Figure 2 (a). In this case, the equation | x | = a has no solutions because the graphs of y = a and y = | x | do not intersect. Copyrighted material. See: 1

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374 Chapter 4 Absolute Value Functions Version: Fall 2007 Case II: a = 0 If a = 0 , then the graph of y = 0 is a horizontal line that coincides with the x -axis, as shown in Figure 2 (b). In this case, the equation | x | = 0 has the single solution x = 0 , because the horizontal line y = 0 intersects the graph of y = | x | at exactly one point, at x = 0 . Case III: a > 0 If a > 0 , then the graph of y = a is a horizontal line that lies strictly above the x - axis, as shown in Figure 2 (c). In this case, the equation | x | = a has two solutions, because the graphs of y = a and y = | x | have two points of intersection. Recall that the left-hand branch of y = | x | has equation y = x , and points on this branch have the form ( 1 , 1) , ( 2 , 2) , etc. Because the point where the graph of y = a intersects the left-hand branch of y = | x | has y -coordinate y = a , the x -coordinate of this point of intersection is x = a . This is one solution of | x | = a . Recall that the right-hand branch of y = | x | has equation y = x , and points on this branch have the form (1 , 1) , (2 , 2) , etc. Because the point where the graph of y = a intersects the right-hand branch of y = | x | has y -coordinate y = a , the x -coordinate of this point of intersection is x = a . This is the second solution of | x | = a . x y y = | x | y = a x y y = | x | y = a 0 x y y = | x | y = a - a a (a) a < 0 . (b) a = 0 . (c) a > 0 . Figure 2. The solution of | x | = a has three cases. This discussion leads to the following key result.
Section 4.3 Absolute Value Equations 375 Version: Fall 2007 Property 2. The solution of | x | = a depends upon the value and sign of a . Case I: a < 0 The equation | x | = a has no solutions. Case II: a = 0 The equation | x | = 0 has one solution, x = 0 . Case III: a > 0 The equation | x | = a has two solutions, x = a or x = a . Let’s look at some examples. l⚏ Example 3. Solve | x | = 3 for x .

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• Negative and non-negative numbers, single solution

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