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### Chapter 12 Section 4

Solving Multi-Step Inequalities

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

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 only exception is that with inequalities, you must reverse the inequality symbol if you multiply or divide by a

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

Your Turn only exception is that with inequalities, you must reverse the inequality symbol if you multiply or divide by a

Solve each inequality. Check your solution.

10 – 5x < 25

{x l x > -3}

Your Turn only exception is that with inequalities, you must reverse the inequality symbol if you multiply or divide by a

Solve each inequality. Check your solution.

3x + 1 > -17

{x l x < 6}

Example 4 only exception is that with inequalities, you must reverse the inequality symbol if you multiply or divide by a

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

Your Turn only exception is that with inequalities, you must reverse the inequality symbol if you multiply or divide by a

Solve each inequality. Check your solution.

2 > -(x + 7)

{x l x > -9}

Your Turn only exception is that with inequalities, you must reverse the inequality symbol if you multiply or divide by a

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 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?

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

Solve 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?

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 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?

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.