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Multi-Step Inequalities: Solving and Checking Solutions

This lesson focuses on solving multi-step inequalities, providing clear examples and necessary checks for solutions. We'll work through problems like 9 + 3x < 27 and observe the steps for isolating x, confirming that solutions must keep the inequality's integrity, especially when multiplying or dividing by a negative number. Practice exercises are included to reinforce the process. You'll also learn how to apply these techniques in real scenarios, including average score calculations for tests. The main takeaway is understanding the solution sets and validating findings.

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Multi-Step Inequalities: Solving and Checking Solutions

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  1. Chapter 12 Section 4 Solving Multi-Step Inequalities

  2. Example 1 Solve 9 + 3x < 27. Check your solution. 9 + 3x < 27 9 - 9 + 3x < 27 - 9 3x < 18 3 3 x < 6

  3. Check Substitute 6 and a number less than 6 into the inequality. Let x = 6 9 + 3(6) < 27 9 + 18 < 27 27 < 27 False, The solution is {x l x < 6}. Let x = 0 9 + 3(0) < 27 9 < 27 True

  4. Your Turn Solve each inequality. Check your solution. 4 + 2x ≤ 12 {x l x ≤ 4}

  5. Your Turn Solve each inequality. Check your solution. 8x - 5 ≥ 11 {x l x ≥ 2}

  6. 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.

  7. 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 x ≤ -2 Your solution is {x l x ≤ -2}. Check your solution. Reverse the symbol

  8. Your Turn Solve each inequality. Check your solution. 10 – 5x < 25 {x l x > -3}

  9. Your Turn Solve each inequality. Check your solution. 3x + 1 > -17 {x l x < 6}

  10. Example 4 Solve 8 ≤ -2(x – 5). Check your solution. 8 ≤ -2(x – 5) 8 ≤ -2x + 10 8 - 10 ≤ -2x + 10 – 10 -2 ≤ -2x -2 -2 1 ≥ x The solution is {x l x ≤ 1). Check your solution. Reverse the symbol

  11. Your Turn Solve each inequality. Check your solution. 2 > -(x + 7) {x l x > -9}

  12. Your Turn Solve each inequality. Check your solution. 3(x – 4) ≤ x - 5 {x l x ≤ 3.5}

  13. 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.

  14. Plan The sum of Hannah’s four test scores, divided by 4, will give the mean score. The mean must be more than 92.

  15. 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

  16. 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.

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