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4.3 Absolute Value Equations 373 Section 4.3 Absolute Value Equations 373 4.3 Absolute Value Equations In the previous section, we defined −x, if x < 0. |x| = (1) x, if x ≥ 0, and we saw that the graph of the absolute value function defined by f(x) = |x| has the “V-shape” shown in Figure 1. y 10 x 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. 1 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ Version: Fall 2007 374 Chapter 4 Absolute Value Functions • 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. y y=|x| y y=|x| y y=|x| y=a y=a x x x 0 −a a y=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. Version: Fall 2007 Section 4.3 Absolute Value Equations 375 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. I Example 3. Solve |x| = −3 for x. The graph of the left-hand side of |x| = −3 is the “V” of Figure 2(a). The graph of the right-hand side of |x| = −3 is a horizontal line three units below the x-axis. This has the form of the sketch in Figure 2(a). The graphs do not intersect. Therefore, the equation |x| = −3 has no solutions. An alternate approach is to consider the fact that the absolute value of x can never equal −3. The absolute value of a number is always nonnegative (either zero or positive). Hence, the equation |x| = −3 has no solutions. I Example 4. Solve |x| = 0 for x. This is the case shown in Figure 2(b). The graph of the left-hand side of |x| = 0 intersects the graph of the right-hand side of |x| = 0 at x = 0. Thus, the only solution of |x| = 0 is x = 0. Thinking about this algebraically instead of graphically, we know that 0 = 0, but there is no other number with an absolute value of zero. So, intuitively, the only solution of |x| = 0 is x = 0. I Example 5. Solve |x| = 4 for x. The graph of the left-hand side of |x| = 4 is the “V” of Figure 2(c). The graph of the right-hand side is a horizontal line 4 units above the x-axis. This has the form of the sketch in Figure 2(c). The graphs intersect at (−4, 4) and (4, 4). Therefore, the solutions of |x| = 4 are x = −4 or x = 4. Alternatively, | − 4| = 4 and |4| = 4, but no other real numbers have absolute value equal to 4. Hence, the only solutions of |x| = 4 are x = −4 or x = 4. Version: Fall 2007 376 Chapter 4 Absolute Value Functions I Example 6. Solve the equation |3 − 2x| = −8 for x. If the equation were |x| = −8, we would not hesitate. The equation |x| = −8 has no solutions. However, the reasoning applied to the simple case |x| = −8 works equally well with the equation |3 − 2x| = −8. The left-hand side of this equation must be nonnegative, so its graph must lie above or on the x-axis. The right-hand side of |3−2x| = −8 is a horizontal line 8 units below the x-axis. The graphs cannot intersect, so there is no solution. We can verify this argument with the graphing calculator. Load the left and right- hand sides of |3 − 2x| = −8 into Y1 and Y2, respectively, as shown in Figure 3(a). Push the MATH button on your calculator, then right-arrow to the NUM menu, as shown in Figure 3(b). Use 1:abs( to enter the absolute value shown in Y1 in Figure 3(a). From the ZOOM menu, select 6:ZStandard to produce the image shown in Figure 3(c). Note, that as predicted above, the graph of y = |3 − 2x| lies on or above the x- axis and the graph of y = −8 lies strictly below the x-axis. Hence, the graphs cannot intersect and the equation |3 − 2x| = −8 has no solutions. (a) (b) (c) Figure 3. Using the graphing calculator to examine the solution of |3 − 2x| = −8. Alternatively, we can provide a completely intuitive solution of |3 − 2x| = −8 by arguing that the left-hand side of this equation is nonnegative, but the right-hand side is negative. This is an impossible situation. Hence, the equation has no solutions. I Example 7. Solve the equation |3 − 2x| = 0 for x. We have argued that the only solution of |x| = 0 is x = 0. Similar reasoning points out that |3 − 2x| = 0 only when 3 − 2x = 0. We solve this equation independently. 3 − 2x = 0 −2x = −3 3 x = 2 Thus, the only solution of |3 − 2x| = 0 is x = 3/2. It is worth pointing out that the “tip” or “vertex” of the “V” in Figure 3(c) is located at x = 3/2. This is the only location where the graphs of y = |3 − 2x| and y = 0 intersect. Version: Fall 2007 Section 4.3 Absolute Value Equations 377 I Example 8. Solve the equation |3 − 2x| = 6 for x. In this example, the graph of y = 6 is a horizontal line that lies 6 units above the x-axis, and the graph of y = |3 − 2x| intersects the graph of y = 6 in two locations. You can use the intersect utility to find the points of intersection of the graphs, as we have in Figure 4(b) and (c). (a) (b) (c) Figure 4. Using the graphing calculator to find two solutions of |3 − 2x| = 6. Expectations. We need a way of summarizing this graphing calculator approach on our homework paper. First, draw a reasonable facsimile of your calculator’s viewing window on your homework paper. Use a ruler to draw all lines. Complete the following checklist. • Label each axis, in this case with x and y. • Scale each axis. To do this, press the WINDOW button on your calculator, then report the values of xmin, xmax, ymin, and ymax on the appropriate axis. • Label each graph with its equation. • Drop dashed vertical lines from the points of intersection to the x-axis. Shade and label these solutions of the equation on the x-axis. Following the guidelines in the above checklist, we obtain the image in Figure 5. y 10 y=|3−2x| y=6 x −10 −1.5 4.5 10 −10 Figure 5. Reporting a graphical solu- tion of |3 − 2x| = 6. Version: Fall 2007 378 Chapter 4 Absolute Value Functions Algebraic Approach. One can also use an algebraic technique to find the two solutions of |3 − 2x| = 6. Much as |x| = 6 has solutions x = −6 or x = 6, the equation |3 − 2x| = 6 is possible only if the expression inside the absolute values is either equal to −6 or 6. Therefore, write 3 − 2x = −6 or 3 − 2x = 6, and solve these equations independently.
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