Name Class Date. Solving an Absolute Value Equation by Graphing. and the second equation y = 5 as Y 2
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1 Name Class Date 1-7 Solving Absolute Value Equations Going Deeper Essential question: How can you use graphing to solve equations involving absolute value? The equation 2 x = 5 is an example of an absolute value equation. 1 A-REI.4.11 EXAMPLE Solving an Absolute Value Equation by Graphing Use a graphing calculator to solve the equation 2 x = 5. A Treat the left side of the equation as the absolute value equation y = 2 x Treat the right side as the equation y = 5. B Press Y=. Enter the first equation, y = 2 x , as Y 1 and the second equation y = 5 as Y 2. C Press GRAPH. Copy the graph from your calculator on the grid at the right. Use the intersect feature under the CALC menu to find the points of intersection of Y 1 and Y 2. D Identify the x-coordinate of each point where the graphs of Y 1 and Y 2 intersect. Show that each x-coordinate is a solution of 2 x = 5. REFLECT 1a. Why is the y-coordinate of both points of intersection equal to 5? 1b. The vertex of an absolute value graph is the lowest point if the graph opens upward or the highest point if the graph opens downward. The vertex of the graph of y = 2 x is (1, 3). How are the coordinates of the vertex related to its equation? Chapter 1 41 Lesson 7
2 2 A-REI.1.1 EXAMPLE Solving an Absolute Value Equation Using Algebra Solve the equation 2 x = 5 using algebra. A Isolate the expression x x = 5 Write the equation. Subtract 1 from both sides. 2 x - 3 = Simplify. 2 x - 3 = Divide both sides by 2. x - 3 = Simplify. B Interpret the equation x - 3 = 2: What numbers have an absolute value equal to 2? C Set the expression inside the absolute value bars equal to each of the numbers from Part B and solve for x. x - 3 = or x - 3 = Write an equation for each value of x - 3. x = or x = Simplify. Add 3 to both sides of each equation. REFLECT 2a. The left side of the equation is 2 x Evaluate this expression for each solution of the equation. How does this help you check the solutions? 2b. Suppose the number on the right side of the equation was -5 instead of 5. What solutions would the equation have? Why? When answering these questions, you may want to refer to the graph of y = 2 x Chapter 1 42 Lesson 7
3 3 A-CED.1.2 EXAMPLE Solving a Real-World Problem Sal exercises by running east 3 miles along a road in front of his home and then reversing his direction to return home. He runs at a constant speed of 0.1 mile per minute. Write and graph an absolute value equation that gives his distance d (in miles) from home in terms of the elapsed time t (in minutes). Use the graph to find the time(s) at which Sal is 1 mile from home. A Determine the three key values of the distance equation: When Sal begins his run (t = 0 minutes), he is miles from home, so d =. When Sal reverses direction, he is miles from home. miles He reaches this point in t = = minutes, so 0.1 mile per minute when t =, d =. When Sal returns home, he is miles from home. Because he has miles run a total of 6 miles, he reaches this point in t = 0.1 mile per minute = minutes, so when t =, d =. B Add axis labels and scales to the coordinate plane shown, then plot the points (t, d) using the time and distance values from part A. The equation is an absolute value equation, and the vertex of the equation s graph is the point that represents when Sal reverses direction. Draw the complete graph and then write the absolute value equation. d = -0.1 t - + C To find the time(s) when Sal is 1 mile from home, draw the graph of y = 1. Find the t-coordinate of each point where the two graphs intersect. REFLECT d 30 t 3a. Show how to use algebra to find the time(s) when Sal is 1 mile from home. Chapter 1 43 Lesson 7
4 PRACTICE Use a graphing calculator to solve each absolute value equation. Sketch your graphs on the grids provided x = x = 4 3. x = -2 Solve each absolute value equation using algebra x = x = x = 8 7. The number of gallons of water in a storage tank is given by y = -40 h where h is the time in hours since the tank was last empty. a. What is the maximum number of gallons the tank can hold? b. For what values of h is the tank half empty? c. For what values of h is the tank empty? d. if the tank is empty, how long does it take to refill it? 8. The number of shoppers in a store is modeled by y = -0.5 t where t is the time (in minutes) since the store opened at 10:00 a.m. a. For what values of t are there 100 shoppers in the store? b. At what times are there 100 shoppers in the store? c. What is the greatest number of shoppers in the store? d. At what time does the greatest number of shoppers occur? Chapter 1 44 Lesson 7
5 Name Class Date Additional Practice 1-7 Solve each equation. 1. x = x = x 6 = x = x = x + 3 = x 1 = x 5 = x = x = x 1 = x = How many solutions does the equation x + 7 = 1 have? 14. How many solutions does the equation x + 7 = 0 have? 15. How many solutions does the equation x + 7 = 1 have? Leticia sets the thermostat in her apartment to 68 degrees. The actual temperature in her apartment can vary from this by as much as 3.5 degrees. 16. Write an absolute-value equation that you can use to find the minimum and maximum temperature. 17. Solve the equation to find the minimum and maximum temperature. Chapter 1 45 Lesson 7
6 Problem Solving Write the correct answer. 1. A machine manufactures wheels with a diameter of 70 cm. It is acceptable for the diameter of a wheel to be within 0.02 cm of this value. Write and solve an absolute-value equation to find the minimum and maximum acceptable diameters. 3. Two numbers on a number line are represented by the absolute-value equation n 5 = 6. What are the two numbers? 2. A pedestrian bridge is 53 meters long. Due to changes in temperature, the bridge may expand or contract by as much as 21 millimeters. Write and solve an absolute-value equation to find the minimum and maximum lengths of the bridge. 4. A jewelry maker cuts pieces of wire to shape into earrings. The equation x 12.2 = 0.3 gives the minimum and maximum acceptable lengths of the wires in centimeters. What is the minimum acceptable length of a wire? The table shows the recommended daily intake of several minerals for adult women. Use the table for questions 5 7. Select the best answer. 5. Which absolute-value equation gives the minimum and maximum recommended intakes for zinc? A x 8 = 32 C x 16 = 24 B x 24 = 16 D x 40 = Jason writes an equation for the minimum and maximum intakes of fluoride. He writes it in the form x b = c. What is the value of b? A 3 C 6.5 B 3.5 D 7 6. For which mineral are the minimum and maximum recommended intakes given by the absolute-value equation x 31.5 = 13.5? Mineral F Fluoride G Iron Daily Minimum (mg) H Zinc J None of these Daily Maximum (mg) Fluoride 3 10 Iron Zinc 8 40 Source: Chapter 1 46 Lesson 7
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