4-9. Solving Absolute Value Equations and Inequalities. Solving x = a. Lesson

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1 Chapter Lesson -9 Solving Absolute Value Equations and Inequalities BIG IDEA Inequalities with a + b on one side and a positive number c on the other side can be solved b using the fact that a + b must equal either c or c. A school carnival had a Guess the Number booth featuring a jar full of pennies. Whoever guessed closest to the actual number of pennies would win a prize. Onl the principal knew there were 7 pennies in the jar. When the prize was announced, the winner was off b 9 pennies. How man pennies did the winner guess? The winning guess deviated from the actual number of pennies b 9. This does not sa if the guess was too high or too low. It could have been either = 1 or 7-9 = 3. These two possibilities are shown on the number line at the right. The two numbers 3 and 1 are the solutions of n - 7 = 9. The epression n - 7 is the absolute deviation of the guess n, from the actual number of pennies, 7. In this case, n - 7 = 9 and the solutions to the equation are 3 and 1. The equation n - 7 = 9 is of the form a + b = c, with a = 1, n in place of, b = 7, and c = 9. All equations of this form can be solved using what ou know about linear equations and compound sentences. Mental Math When a > 0, determine if the following are positive or negative. a. ( a ) b. ( 5a) 3 c. ( 0.9a) n 7 Solving = a Remember that is the distance of from 0 on a number line. At the right is a table of some pairs of values of and and the graph of =. Also, the line = 3 is graphed ( 3, 3) 3 (0, 0) (, ) (3, 3) 3 More Linear Equations and Inequalities

2 Lesson -9 The following conclusions about the equation = c can be made from the table and graph. is never negative, so there are no solutions to = or to an other equation of the form = c when c is negative. = 0 onl when = 0. = 3 has two solutions, 3 and 3. The two solutions can be seen in the table in the rows where = 3. Also, the graph of the horizontal line = 3 intersects = in two points, where = 3 and = 3. Therefore, there are alwas two solutions to = c when c is positive. Solutions to = c When c is positive, then there are two solutions to = c, namel c and c. When c is negative, then there are no solutions to = c. When c is zero, then = 0 and there is one solution: = 0. QY1 Solving a + b = c The above ideas appl to an equation where there is an absolute value of an epression on one side and a number on the other. QY1 Find all the solutions to each equation. a. t = 15 b. u =. c. v = 0 GUIDED Eample 1 Consider the graph, which shows 1 = - and = 1. Use the graph and use algebraic properties to solve - = 1. Solution 1 Use the graph. The points of intersection of the two graphs are? and?. The -coordinates of these points of intersection are? and?. The solutions to - = 1 are? and?. Solution Use algebraic properties. Ask ourself: What numbers have absolute value 1? - =? or - =? Solve this compound sentence as ou did in Lesson -. = +? or = +?? = or? =? = or? = You should have the same answers from both Solutions 1 and Solving Absolute Value Equations and Inequalities

3 Chapter Solving < c and > c When c is positive, the two solutions to = c can be represented on a number line. The are the points at a distance c from 0. = c if and onl if = c or = c. The points closer to 0 than c are the solutions to the inequalit < c. The can be described b the interval c < < c. For eample, the solutions to < 3 can be described b the double inequalit 3 < < 3. < c if and onl if c < < c. The solutions to the inequalit > 3 are the points whose distance from 0 is greater than 3. The graph of these points has two parts and is described b the compound inequalit < 3 or > 3. > c if and onl if < c or > c. Solutions to < c when c is positive < c if and onl if c < < c. Solutions to > c when c is positive > c if and onl if < c or > c. c c c 0 c 0 c 0 c QY Solving a + b < c, a + b > c To solve these inequalities, think of the simpler inequalities > c and < c. QY Describe all solutions to d > 0.9 using a compound inequalit without the absolute value smbol. GUIDED Eample Solve + 0. Solution 1 Think: < a means a < < a. + 0 Write the inequalit < a means a < < a.?? Add to each side.?? Divide both sides b. 3 More Linear Equations and Inequalities

4 Lesson -9 Solution Use a graph or table. Ask ourself: When is + below or at 0 on the graph? When is + less than or equal to 0 in the table? +????? 0???? ; scl = 1 0 0; scl = So the solution to + 0 is??. Questions COVERING THE IDEAS 1. Multiple Choice You are asked for the ear of the Emancipation Proclamation in the United States on a test. The correct answer is 13. You guessed g and ou were off b ears. What equation s solution gives the possible values of g? A 13 - = g B g = 13 - C g - 13 = D g - = 13. Determine whether the number is a solution to the equation 0 = n a. 30 b. 30 c. 150 d. 150 In 3, find all solutions in our head. 3. A =. B = C = 0. 5 D = 0 7. Use the table at the right to solve each sentence. a. - 3 = 7 b. - 3 < 7 c. - 3 > 7 Y 1 = - 3 Y = In September 1, Abraham Lincoln called on the seceded states to return to the Union or have their slaves declared free. When no state returned, he issued the proclamation on Januar 1, 13. Source: Britannica Solving Absolute Value Equations and Inequalities 37

5 Chapter. Fill in the Blanks The sentence a + b = 15 is equivalent to the compound sentence? or?. 9. Multiple Choice The green portion of the graph can be used to find the solution to which of the following? A + 5 = B + 5 C + 5 D + 5 >. Fill in the Blanks - =.3 means the distance between? and? on a number line is? In 11 1, solve the sentence. 11. a + = b = c < 9 1. d > g h Use the graph of = 7 - at the right to solve the sentence. a. 7 - = 9 b. 7 - = 0 c. 7 - = d. 7 - > 5 APPLYING THE MATHEMATICS 1. Let 5 + = m. Find a value of m so that the absolute value equation has the given number of solutions. a. two solutions b. one solution c. no solutions 19. A bo of Wheat-Os breakfast cereal sas that it contains ounces. However, because the machiner that fills the boes cannot be eactl precise, the can be from 1 ounce below to 1 ounce above this weight. a. Graph the possible number of ounces of Wheat-Os. b. Write an absolute value inequalit to show this amount. 0. It is recommended that teenagers get.5 ± 0.7 hours of sleep. a. Write a double inequalit to epress the recommended amount of sleep. b. Write an absolute value inequalit for the recommended amount of sleep. 1. Write a single inequalit for the graph below More Linear Equations and Inequalities

6 . At the right is the graph that Zoe used to solve < + 1. Use her graph and intersection points to give the solution. REVIEW 3. Because of traffic, Nina s drive to school takes anwhere from 1 to 0 minutes. (Lesson -) a. Write an inequalit epressing the time it takes Nina to get to school. b. Write an epression of the form a ± d to represent the interval in Part a. In and 5, solve the compound inequalit and graph the solution. (Lesson -). a and 1 - a > 5a (m - ) 7m + or m - 7 > 30. Consider the equations = and 3 - = a -. Find a value of a so that the two lines do not intersect. (Lesson -7) 7. Write equations for the horizontal and vertical lines that go through the point (.5, 9 11 ). (Lesson -). Gail had $11.50 to spend on snacks for herself and her friends. She wanted to bu as man 5 cent energ bars as she could, in addition to 3 boes of popcorn at 75 cents each and 3 juice bottles at $1.50 each. Set up an inequalit and solve b clearing decimals to find the number of energ bars she can bu. (Lesson 3-) EXPLORATION 9. You learned that absolute value equations can have 0, 1, or solutions. a. Write an absolute value inequalit whose solution is all real numbers. b. Write an absolute value inequalit that has no solutions. 30. Solve - = -. Lesson -9 (7, ) ( 1, 0) QY ANSWERS 1a. t = 15 or t = 15 b. no solution c. v = 0. d < 0.9 or d > 0.9 Solving Absolute Value Equations and Inequalities 39