Question

Solve each compound inequality. Write your answers in interval notation.

8. 6p+8 <8+10p or 4+6p>7p+5 answer should be: p≥0 or <-1 (-infinity -1) U [0, infinity)
9. 8-8x<-6+6x or 10x+5<2+9x answer should be: (- infinity, -3) U ( 1, infinity)
10. -6b-8>-7b+1 or 4b+2 ≥10+8b answer should be: (- infinity , -2] U (9, infinity)

Answer 1

Let's solve each compound inequality step by step and present the answers in interval notation.

### Problem 8:
Given inequalities:
1. \( 6p + 8 < 8 + 10p \)
2. \( 4 + 6p > 7p + 5 \)

**1. Solve \( 6p + 8 < 8 + 10p \):**
\[
6p + 8 < 8 + 10p \\
6p - 10p < 8 - 8 \\
-4p < 0 \\
p > 0
\]

**2. Solve \( 4 + 6p > 7p + 5 \):**
\[
4 + 6p > 7p + 5 \\
4 - 5 > 7p - 6p \\
-1 > p \\
p < -1
\]

**Combine the solutions:**
The solution set is:
\[ p < -1 \quad \text{or} \quad p > 0 \]
In interval notation, this is:
\[ (-\infty, -1) \cup (0, \infty) \]

### Problem 9:
Given inequalities:
1. \( 8 - 8x < -6 + 6x \)
2. \( 10x + 5 < 2 + 9x \)

**1. Solve \( 8 - 8x < -6 + 6x \):**
\[
8 + 6 < 8x + 6x \\
14 < 14x \\
x > 1
\]

**2. Solve \( 10x + 5 < 2 + 9x \):**
\[
10x - 9x < 2 - 5 \\
x < -3
\]

**Combine the solutions:**
The solution set is:
\[ x < -3 \quad \text{or} \quad x > 1 \]
In interval notation, this is:
\[ (-\infty, -3) \cup (1, \infty) \]

### Problem 10:
Given inequalities:
1. \( -6b - 8 > -7b + 1 \)
2. \( 4b + 2 \geq 10 + 8b \)

**1. Solve \( -6b - 8 > -7b + 1 \):**
\[
b - 8 > 1 \\
b > 9
\]

**2. Solve \( 4b + 2 \geq 10 + 8b \):**
\[
4b - 8b \geq 10 - 2 \\
-4b \geq 8 \\
b \leq -2
\]

**Combine the solutions:**
The solution set is:
\[ b \leq -2 \quad \text{or} \quad b > 9 \]
In interval notation, this is:
\[ (-\infty, -2] \cup (9, \infty) \]

### Summary of Answers:
- **Problem 8:** \((- \infty, -1) \cup (0, \infty)\)
- **Problem 9:** \((- \infty, -3) \cup (1, \infty)\)
- **Problem 10:** \((- \infty, -2] \cup (9, \infty)\)