https://file.aiquickdraw.com/mathbot/file/1726063115701bgz2bauf.jpg

The image contains four inequalities, and each inequality has a corresponding number line representation. I'll solve each inequality and provide the number line representation based on the given intervals.

### 1. \( 7x + 8 < 29 \)

**Step 1: Solve for \( x \)**
\[
7x + 8 < 29
\]
Subtract 8 from both sides:
\[
7x < 21
\]
Divide both sides by 7:
\[
x < 3
\]

**Number Line Representation:**
- The solution is \( x < 3 \), meaning all values to the left of 3.
- The point at 3 will be an open circle because the inequality is strict (less than, but not equal to).

### 2. \( 2x - 1 > 5 \)

**Step 1: Solve for \( x \)**
\[
2x - 1 > 5
\]
Add 1 to both sides:
\[
2x > 6
\]
Divide both sides by 2:
\[
x > 3
\]

**Number Line Representation:**
- The solution is \( x > 3 \), meaning all values to the right of 3.
- The point at 3 will be an open circle since it's a strict inequality (greater than, but not equal to).

### 3. \( 32 - x \geq 29 \)

**Step 1: Solve for \( x \)**
\[
32 - x \geq 29
\]
Subtract 29 from both sides:
\[
3 \geq x
\]
Or equivalently:
\[
x \leq 3
\]

**Number Line Representation:**
- The solution is \( x \leq 3 \), meaning all values to the left of 3.
- The point at 3 will be a closed circle because the inequality includes "equal to."

### 4. \( 33 - 8x \leq 9 \)

**Step 1: Solve for \( x \)**
\[
33 - 8x \leq 9
\]
Subtract 33 from both sides:
\[
-8x \leq -24
\]
Divide both sides by -8 (remember to flip the inequality sign when dividing by a negative number):
\[
x \geq 3
\]

**Number Line Representation:**
- The solution is \( x \geq 3 \), meaning all values to the right of 3.
- The point at 3 will be a closed circle since the inequality includes "equal to."

---

These are the solutions along with the corresponding number line representations.

Would you like me to clarify any of the steps, or do you have any specific questions about one of the inequalities?

Here are five related questions to help expand your understanding:

1. How do strict inequalities differ from non-strict inequalities when representing them on a number line?
2. Why does dividing or multiplying by a negative number flip the inequality sign?
3. What happens if an inequality involves fractions? How do we solve it?
4. How can inequalities be combined or solved simultaneously?
5. How does solving an absolute value inequality differ from regular inequalities?

**Tip:** When solving inequalities, always double-check whether you need to flip the inequality sign, especially when multiplying or dividing by negative numbers.

Solve the inequalities: 7x + 8 < 29, 2x - 1 > 5, 32 - x ≥ 29, 33 - 8x ≤ 9.

Learn how to solve linear inequalities like 7x + 8 < 29 and graph them on a number line. This solution covers key algebraic techniques for solving inequalities and graphing their solutions. Suitable for students in Grades 8-10.

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