Answer:
Let's solve and represent the solutions on a number line for each inequality.
h) |3x - 8| > 7
Step 1: Set up two cases based on the absolute value:
Case 1: 3x - 8 > 7
3x - 8 - 8
3x > 15
x > 15/3
x > 5
Case 2: -(3x - 8) > 7
-3x + 8 > 7
-3x > 7 - 8
-3x > -1
x < (-1)/(-3)
x < 1/3
So, the solutions to the inequality |3x - 8| > 7 are x > 5 and x < 1/3. Now, let's represent these solutions on a number line:
```
-∞---(1/3)---(5)---∞
x < 1/3 x > 5
```
i) 5|2x + 1| - 3 ≥ 9
Step 1: Add 3 to both sides of the inequality:
5|2x + 1| ≥ 12
Step 2: Divide both sides by 5:
|2x + 1| ≥ 12/5
Step 3: Set up two cases based on the absolute value:
Case 1: 2x + 1 ≥ 12/5
2x + 1 - 1 ≥ 12/5 - 1
2x ≥ 7/5
x ≥ (7/5) * (1/2)
x ≥ 7/10
Case 2: -(2x + 1) ≥ 12/5
-2x - 1 ≥ 12/5
-2x - 1 + 1 ≥ 12/5 + 1
-2x ≥ 17/5
x ≤ (17/5) * (-1/2)
x ≤ -17/10
So, the solutions to the inequality 5|2x + 1| - 3 ≥ 9 are x ≥ 7/10 and x ≤ -17/10. Now, let's represent these solutions on a number line:
```
-∞---(-17/10)---(7/10)---∞
x ≤ -17/10 x ≥ 7/10
```