Question

Solve and fill the number line *SHOW YOUR WORK*

h) |3x-8|>7

i) 5|2x+1| -3 ≥ 9

Answer 1

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

```