2.3 Solving Absolute Value Inequalities
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1 .3 Solving Absolute Value Inequalities Essential Question: What are two was to solve an absolute value inequalit? Resource Locker Eplore Visualizing the Solution Set of an Absolute Value Inequalit You know that when solving an absolute value equation, it s possible to get two solutions. Here, ou will eplore what happens when ou solve absolute value inequalities. Determine whether each of the integers from -5 to 5 is a solution of the inequalit + < 5. If a number is a solution, plot it on a number line. Determine whether each of the integers from -5 to 5 is a solution of the inequalit + > 5. If a number is a solution, plot it on a number line. State the solutions of the equation + = 5 and relate them to the solutions ou found for the inequalities in Steps A and B. If is an real number and not just an integer, graph the solutions of + < 5 and + > 5. Reflect 1. It s possible to describe the solutions of + < 5 and + > 5 using inequalities that don t involve absolute value. For instance, ou can write the solutions of + < 5 as > -3 and < 3. Notice that the word and is used because must be both greater than -3 and less than 3. How would ou write the solutions of + > 5? Eplain.. Describe the solutions of + 5 and + 5 using inequalities that don t involve absolute value. Module 63 Lesson 3
2 Eplain 1 Solving Absolute Value Inequalities Graphicall You can use a graph to solve an absolute value inequalit of the form ƒ () > g () or ƒ () < g (), where ƒ () is an absolute value function and g () is a constant function. Graph each function separatel on the same coordinate plane and determine the intervals on the -ais where one graph lies above or below the other. For ƒ () > g (), ou want to find the -values for which the graph ƒ () is above the graph of g (). For ƒ () < g (), ou want to find the -values for which the graph of ƒ () is below the graph of g (). Eample 1 Solve the inequalit graphicall > The inequalit is of the form ƒ () > g (), so determine the intervals on the -ais where the graph of ƒ () = lies above the graph of g () = The graph of ƒ () = lies above the graph of g () = to the left of = -6 and to the right of = 0, so the solution of > is < -6 or > < 1 The inequalit is of the form f () < g (), so determine the intervals on the -ais where the graph of f () = lies below the graph of g () = 1. The graph of f () = lies below the graph of g () = 1 between = - and = 6, so the solution of < 1 is > - and < Reflect 3. Suppose the inequalit in Part A is instead of >. How does the solution change?. In Part B, what is another wa to write the solution > - and < 6? 5. Discussion Suppose the graph of an absolute value function ƒ () lies entirel above the graph of the constant function g (). What is the solution of the inequalit ƒ () > g ()? What is the solution of the inequalit ƒ () < g ()? Your Turn 6. Solve graphicall. Module 6 Lesson 3
3 Eplain Solving Absolute Value Inequalities Algebraicall To solve an absolute value inequalit algebraicall, start b isolating the absolute value epression. When the absolute value epression is b itself on one side of the inequalit, appl one of the following rules to finish solving the inequalit for the variable. Solving Absolute Value Inequalities Algebraicall 1. If > a where a is a positive number, then < -a or > a.. If < a where a is a positive number, then -a < < a. Eample Solve the inequalit algebraicall. Graph the solution on a number line > 1 - > 6 - < -6 or - > 6 - < -10 or - > > 10 or < - The solution is > 10 or < and + -1 and The solution is -1 and, or Reflect 7. In Part A, suppose the inequalit were > 1 instead of > 1. How would the solution change? Eplain. 8. In Part B, suppose the inequalit were instead of How would the solution change? Eplain. Your Turn Solve the inequalit algebraicall. Graph the solution on a number line < 5 Module 65 Lesson 3
4 Eplain 3 Solving a Real-World Problem with Absolute Value Inequalities Absolute value inequalities are often used to model real-world situations involving a margin of error or tolerance. Tolerance is the allowable amount of variation in a quantit. Eample 3 A machine at a lumber mill cuts boards that are 3.5 meters long. It is acceptable for the length to differ from this value b at most 0.0 meters. Write and solve an absolute value inequalit to find the range of acceptable lengths. Analze Information Identif the important information. The boards being cut are 3.5 meters long. The length can differ b at most 0.0 meters. Formulate a Plan Let the length of a board be l. Since the sign of the difference between l and 3.5 doesn t matter, take the absolute value of the difference. Since the absolute value of the difference can be at most 0.0, the inequalit that models the situation is l Solve l l and l l 3.3 and l 3.7 So, the range of acceptable lengths is 3.3 l 3.7. Justif and Evaluate The bounds of the range are positive and close to 3.5, so this is a reasonable answer. The answer is correct since = 3.5 and = 3.5. Your Turn 11. A bo of cereal is supposed to weigh 13.8 oz, but it s acceptable for the weight to var as much as 0.1 oz. Write and solve an absolute value inequalit to find the range of acceptable weights. Module 66 Lesson 3
5 Elaborate 1. Describe the values of that satisf the inequalities < a and > a where a is a positive constant. 13. How do ou algebraicall solve an absolute value inequalit? 1. Eplain wh the solution of > a is all real numbers if a is a negative number. 15. Essential Question Check-In How do ou solve an absolute value inequalit graphicall? Evaluate: Homework and Practice 1. Determine whether each of the integers from -5 to 5 is a solution of the inequalit If a number is a solution, plot it on a number line.. Determine whether each of the integers from -5 to 5 is a solution of the inequalit If a number is a solution, plot it on a number line. Solve each inequalit graphicall > _ + < Match each graph with the corresponding absolute value inequalit. Then give the solution of the inequalit. A. + 1 > 3 B. + 1 < 3 C. - 1 > 3 D. - 1 < Module 67 Lesson 3
6 Solve each absolute value inequalit algebraicall. Graph the solution on a number line _ > < < < Solve each problem using an absolute value inequalit. 17. The thermostat for a house is set to 68 F, but the actual temperature ma var b as much as F. What is the range of possible temperatures? 18. The balance of Jason s checking account is $30. The balance varies b as much as $80 each week. What are the possible balances of Jason s account? 19. On average, a squirrel lives to be 6.5 ears old. The lifespan of a squirrel ma var b as much as 1.5 ears. What is the range of ages that a squirrel lives? 0. You are plaing a histor quiz game where ou must give the ears of historical events. In order to score an points at all for a question about the ear in which a man first stepped on the moon, our answer must be no more than 3 ears awa from the correct answer, What is the range of answers that allow ou to score points? Module 68 Lesson 3
7 1. The speed limit on a road is 30 miles per hour. Drivers on this road tpicall var their speed around the limit b as much as 5 miles per hour. What is the range of tpical speeds on this road? H.O.T. Focus on Higher Order Thinking. Represent Real-World Problems A poll of likel voters shows that the incumbent will get 51% of the vote in an upcoming election. Based on the number of voters polled, the results of the poll could be off b as much as 3 percentage points. What does this mean for the incumbent? 3. Eplain the Error A student solved the inequalit > 1 graphicall. Identif and correct the student s error. I graphed the functions ƒ() = and g() = 1. Because the graph of g() lies above the graph of ƒ() between = -3 and = 5, the solution of the inequalit is -3 < < Multi-Step Recall that a literal equation or inequalit is one in which the constants have been replaced b letters. a. Solve a + b > c for. Write the solution in terms of a, b, and c. Assume that a > 0 and c 0. b. Use the solution of the literal inequalit to find the solution of > 1. c. In Part a, eplain how the restrictions a > 0 and c 0 affect finding the solutions of the inequalit. Module 69 Lesson 3
8 Lesson Performance Task The distance between the Sun and each planet in our solar sstem varies because the planets travel in elliptical orbits around the Sun. Here is a table of the average distance and the variation in the distance for the five innermost planets in our solar sstem. Average Distance Variation Earth Mars a. Write and solve an inequalit to represent the range of distances that can occur between the Sun and each planet. b. Calculate the percentage variation (variation divided b average distance) in the orbit of each of the planets. Based on these percentages, which planet has the most elliptical orbit? JPL/NASA Module 70 Lesson 3
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