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Chapter 12 Section 4 Solving Multi-Step Inequalities Example 1 Solve 9 + 3x < 27. Check your solution. 9 + 3x < 27 9 - 9 + 3x < 27 - 9 3x < 18 3 3 x<6 Check Substitute 6 and a number less than 6 into the inequality. Let x = 0 Let x = 6 9 + 3(0) < 27 9 + 3(6) < 27 9 < 27 9 + 18 < 27 True 27 < 27 False, The solution is {x l x < 6}. Your Turn Solve each inequality. Check your solution. 4 + 2x ≤ 12 {x l x ≤ 4} Your Turn Solve each inequality. Check your solution. 8x - 5 ≥ 11 {x l x ≥ 2} Solving inequalities is similar to solving equations. The only exception is that with inequalities, you must reverse the inequality symbol if you multiply or divide by a negitive number. Example 3 Solve -4x + 3 ≥ 23 + 6x. Check your solution. -4x + 3 ≥ 23 + 6x -4x – 6x + 3 ≥ 23 + 6x – 6x -10x + 3 ≥ 23 -10x + 3 - 3 ≥ 23 – 3 -10x ≥ 20 -10 -10 Reverse the symbol x ≤ -2 Your solution is {x l x ≤ -2}. Check your solution. Your Turn Solve each inequality. Check your solution. 10 – 5x < 25 {x l x > -3} Your Turn Solve each inequality. Check your solution. 3x + 1 > -17 {x l x < 6} Example 4 Solve 8 ≤ -2(x – 5). Check your solution. 8 ≤ -2(x – 5) 8 ≤ -2x + 10 8 - 10 ≤ -2x + 10 – 10 -2 ≤ -2x Reverse the symbol -2 -2 1≥x The solution is {x l x ≤ 1). Check your solution. Your Turn Solve each inequality. Check your solution. 2 > -(x + 7) {x l x > -9} Your Turn Solve each inequality. Check your solution. 3(x – 4) ≤ x - 5 {x l x ≤ 3.5} Hannah’s scores on the first three of four 100 point tests were 85, 92, and 90. What score must she receive on the fourth test to have a mean score of more than 92 for all tests? Explore Let s = Hannah’s score on the fourth test. The sum of Hannah’s four test scores, divided by 4, will give the mean score. The mean must be more than 92. Plan The sum of Hannah’s four test scores, divided by 4, will give the mean score. The mean must be more than 92. Solve 85 + 92+ 95+ s > 92 4 4 (85 + 92+ 95+ s ) > 4(92) 4 85 + 92+ 95+ s > 368 267 - 267 + s > 368 - 267 s > 101 Examine Substitute a number greater than 101, such as 102, into the original problem. Hannah’s average would be 92.25. Since 92.25 > 92 is a true statement, the solution is correct. Hannah’s must score more than 101 points out of a 100 point test. Without extra credit, this is not possible. So, Hannah cannot have a mean over 92.