6.5 Solving Linear Inequalities 335 Example 10 Solving a Compound Inequality Solve the compound inequality for x, and graph the solution set. x 4 3 2 5 − < + ≤ Solution To solve this compound inequality, we must isolate the x. To do so, we use the same principles used to solve inequalities. − < + ≤ − < ⎛ + ⎝⎜ ⎞ ⎠⎟ ≤ − < + ≤ − − < + − ≤ − − < ≤ x x x x x 4 3 2 5 2( 4) 2 3 2 2(5) 8 3 10 8 3 3 3 10 3 11 7 Multiply each part of the inequality by 2. Subtract 3 from each part of the inequality. The solution set is graphed on the number line as follows. 211 0 7 211 , x # 7 7 Now try Exercise 43 Example 11 Average Grade A student must have an average, or mean, on five tests that is greater than or equal to 80% but less than 90% to receive a final grade of B. Devon’s grades on the first four tests were 98%, 76%, 86%, and 92%. What range of grades on the fifth test would give him a B in the course? Solution The unknown quantity is the range of grades on the fifth test. First construct an inequality that can be used to determine the range of grades on the fifth test. The average (mean) is determined by adding the grades and dividing the sum by the number of tests. Let x the = fifth grade. Then x Average 98 76 86 92 5 = + + + + For Devon to obtain a B in this course, his average must be greater than or equal to 80 but less than 90. x x x x x x 80 98 76 86 92 5 90 80 352 5 90 5(80) 5 352 5 5(90) 400 352 450 400 352 352 352 450 352 48 98 ≤ + + + + < ≤ + < ≤ ⎛ + ⎝⎜ ⎞ ⎠⎟ < ≤ + < − ≤ + − < − ≤ < Multiply each part of the inequality by 5. Subtract 352 from each part of the inequality Thus, a grade greater than or equal to 48% and less than 98% on the fifth test will result in Devon getting a grade of B in this course. 7 Now try Exercise 57 “In mathematics, the art of posing problems is easier than that of solving them.” Georg Cantor Instructor Resources for Section 6.5 in MyLab Math • Objective-Level Videos 6.5 • PowerPoint Lecture Slides 6.5 • MyLab Exercises and Assignments 6.5
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