To graph, draw a solid circle at 4 and an arrow extending to the right and a solid circle at 5 and an arrow extending to the left.
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1 Solve each inequality. Graph the solution set on a number line. 2. The solution of the inequality is. To graph, draw a solid circle at 1.5 and a solid circle at 4 and draw a line to connect the circles. 4. or The solution of the inequality is. To graph, draw a solid circle at 4 and an arrow extending to the right and a solid circle at 5 and an arrow extending to the left. esolutions Manual - Powered by Cognero Page 1
2 6. The absolute value of a number is always non-negative. So, the inequality is true for any value of q. The solution of the absolute value inequality is. Since the solution set includes all real numbers, draw an arrow extending in each direction. 8. The absolute value of a number is always non-negative. So, no value of x satisfies the inequality. The solution set is. Since there is no solution, leave the graph blank. 10. The solution of the inequality is. To graph, draw a solid circle at -1 and a solid circle at 2.5 and draw a line to connect the circles. esolutions Manual - Powered by Cognero Page 2
3 Solve each inequality. Graph the solution set on a number line. 12. The solution of the inequality is. To graph, draw an open circle at 6 and an open circle at 10 and draw a line to connect the circles. 14. or The solution of the inequality is. To graph, draw an open circle at extending to the right. and an arrow extending to the left and an open circle at 3 and an arrow esolutions Manual - Powered by Cognero Page 3
4 16. The solution of the inequality is. To graph, draw an open circle at 2 and an open circle at 2 and draw a line to connect the circles. 18. or The solution of the inequality is. To graph, draw an open circle at 2 and an arrow extending to the left and an open circle at extending to the right. and an arrow esolutions Manual - Powered by Cognero Page 4
5 20. The solution of the inequality is. To graph, draw an open circle at -1 and an open circle at and draw a line to connect the circles. 22. ANATOMY Forensic scientists use the equation h = 2.6f to estimate the height h of a woman given the length in centimeters f of her femur bone. a. Suppose the equation has a margin of error of ±3 centimeters. Write an inequality to represent the height of a woman given the length of her femur bone. b. If the length of a female skeleton s femur is 50 centimeters, write and solve an absolute value inequality that describes the woman s height in centimeters. a. The margin of error is cm. So: b. Substitute f = 50 in the equation h = 2.6f h = 2.6(50) = The margin of error is cm. So: The woman s height ranges from cm to cm. esolutions Manual - Powered by Cognero Page 5
6 24. Write an absolute value inequality for each graph. Write a compound inequality from the graph. Then determine what can be added or subtracted from each term so that it is in the form x < ÿa or x > a. This implies that x + c > a. or This implies: 26. Write a compound inequality from the graph. Then determine what can be added or subtracted from each term so that it is in the form x < -a or x > a. This implies that x + c > a. This implies: or 28. Write a compound inequality from the graph. Then determine what can be added or subtracted from each term so that it is in the form -a < x + c < a. This implies that x + c < a. This implies: esolutions Manual - Powered by Cognero Page 6
7 30. Write a compound inequality from the graph. Then determine what can be added or subtracted from each term so that it is in the form a < x + c < a. This implies that x + c < a. This implies: 32. GEOMETRY The Exterior Angle Inequality Theorem states that an exterior angle measure is greater than the measure of either of its corresponding remote interior angles. Write two inequalities to express the relationships among the measures of the angles of ABC. Angle 4 is the exterior angle, and angles 1 and 2 are its corresponding remote interior angles. By the Exterior Angle Inequality Theorem: Solve each inequality. Graph the solution set on a number line. 34. or The solution set is. To graph, draw an open circle at 12 and an arrow extending to the left and an open circle at 5 and an arrow extending to the right. esolutions Manual - Powered by Cognero Page 7
8 36. The absolute value of a number is always non-negative. So, no value of k satisfies the inequality. The solution set is. Since there is no solution, leave the graph blank. 38. The solution set is: To graph, draw an open circle at and an open circle at and draw a line to connect the circles. esolutions Manual - Powered by Cognero Page 8
9 40. or The solution set is: To graph, draw a solid circle at and an arrow extending to the left and a solid circle at and an arrow extending to the right. Write an algebraic expression to represent each verbal expression. 42. numbers that are no more than unit from 1 Let x represent the numbers that are no more than unit from 1. So: 44. AUTO RACING NASCAR rules stipulate that a car must conform to a set of 32 templates, each shaped to fit a different contour of the car. When a template is placed on a car, the gap between it and the car cannot exceed the specified tolerance. Each template is marked on its edge with a colored line that indicates the tolerance for the template. esolutions Manual - Powered by Cognero Page 9
10 a. Suppose a certain template is inches long. Use the information in the table at the right to write an absolute value inequality for templates with each line color. b. Find the acceptable lengths for that part of a car if the template has each line color. c. Graph the solution set for each line color on a number line. d. The tolerance of which line color includes the tolerances of the other line colors? Explain your reasoning. a. Let x represent the length for the part of a car. Red: Blue: Green: b. The acceptable length for the part of the car if the template has red line color: The acceptable length for the part of the car if the template has blue line color: The acceptable length for the part of the car if the template has blue line color: c. Red: Blue: Green: esolutions Manual - Powered by Cognero Page 10
11 d. Red. The red line color has the smallest tolerance, 0.07 < 0.25 < 0.5. So, the other line colors would be well within their tolerances. Solve each inequality. Graph the solution set on a number line. 46. and This implies: y > 5 and y < 1 No value of y satisfies the compound inequality. So the solution set is. Since there is no solution leave the graph blank. esolutions Manual - Powered by Cognero Page 11
12 48. The solution set is. To graph, draw a solid circle at 8 and an open circle at 10 and draw a line to connect the circles. 50. and That is, 1 < t < 3. The solution set is. To graph, draw an open circle at 1 and an open circle at 3 and draw a line to connect the circles. esolutions Manual - Powered by Cognero Page 12
13 52. AIR TRAVEL The airline on which Drew is flying has weight restrictions for checked baggage. Drew is checking one bag. a. Describe the range of weights that would classify Drew s bag as free, $25, $50, and unacceptable. b. If Drew s bag weighs 68 pounds, how much will he pay to take it on the plane? a. Let x represent the weight of the bag. There is no charge for the checked baggage if. To cost $25, x should lie between 50 and 70. That is,. To cost 50, x should lie between 50 and 100. That is,. If the baggage is not acceptable, then x should be greater than 100. That is,. b. x = 68. So, the cost for the baggage would be $25. esolutions Manual - Powered by Cognero Page 13
14 54. CHALLENGE Solve Use the Alternate Triangle Inequality. Case 1: When x 0 and x 4: Substitute the test value for x = 1. Therefore, x cannot be positive. Case 2: When x < 0: Substitute the test value for x = 1. So, the solution is x < 0. REASONING Determine whether each statement is true or false. If false, provide a counterexample. 56. The graph of a compound inequality involving an or statement contains a region of values that are not solutions. False; sample answer: the graph of x > 2 or x < 3 includes the entire number line. esolutions Manual - Powered by Cognero Page 14
15 58. WRITING IN MATH An alternate definition of absolute value is to define as the distance between a and b on the number line. Explain how this definition can be used to solve inequalities of the form. Sample answer: represents the distance between some unknown value of the variable x and a point c on the number line. The solution set of the inequality is the set of all numbers such that the distance from the numbers to c is less than r units. Use a number line to find the numbers that are r units from c in either direction. esolutions Manual - Powered by Cognero Page 15
16 60. OPEN ENDED Write an absolute value inequality with a solution of Sample answer: first graph a and b on a number line. The solution of the inequality is the set of points between a and b, inclusive. Mark the midpoint. The distance between points a or b and the center is. So the absolute value inequality will be be. Check this inequality.. Since the absolute value inequality had the solution, the right side of the inequality must or Therefore the inequality is. esolutions Manual - Powered by Cognero Page 16
17 62. WRITING IN MATH Summarize the difference between and compound inequalities and or compound inequalities. Sample answer: A compound inequality that contains and is true if and only if both individual inequalities are true, for example, 3 < x < 1, while an inequality containing or only needs one of the individual inequalities to be true, for example, x < 3 or x > SAT/ACT Find an expression equivalent to F G H J K The correct choice is K. esolutions Manual - Powered by Cognero Page 17
18 66. Which graph represents the solution set for A B C D The correct choice is A. 68. TRAVEL Maggie is planning a 5-day trip to a family reunion. She wants to spend no more than $1000. Her plane ticket is $375, and the hotel is $85 per night. a. Let f represent the cost of food for one day. Write an inequality to represent this situation. b. Solve the inequality and interpret the solution. a. The cost of the food for one day is $f. For 5 days, the cost of the food is $5f. The plane ticket costs $375. For a 5-day stay in the hotel, it costs $425. Total cost for a 5-day trip = $5f + $375 + $425 = f. Maggie wants to spend no more than $1000. Therefore: b. That is, Maggie can spend no more than $40 per day on food. esolutions Manual - Powered by Cognero Page 18
19 Solve each equation. Check your solutions. 70. or Substitute the values in the equation. The solution set is. Name the property illustrated by each statement. 72. If then Addition Property of Equality; the Addition Property of Equality states that for any real numbers a, b, and c, if a = b, then a + c = b + c. 74. If then Associative Property of Addition; the Associate Property of Addition states that the way in which the terms are grouped does not affect the sum. esolutions Manual - Powered by Cognero Page 19
7-2 Solving Exponential Equations and Inequalities. Solve each equation. 1. 3 5x = 27 2x 4 SOLUTION:
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