https://file.aiquickdraw.com/mathbot/file/1727191278840ngb0eql8.jpg

Let's work through each inequality one by one, solve them, and explain the steps involved. Afterward, I can assist with graphing the solutions as well.

### 1. \( b + 6 < 14 \)
- Subtract 6 from both sides:
  \[
  b < 14 - 6
  \]
  \[
  b < 8
  \]

### 2. \( 12 - d > -8 \)
- Subtract 12 from both sides:
  \[
  -d > -8 - 12
  \]
  \[
  -d > -20
  \]
- Multiply by \(-1\) (remember to reverse the inequality):
  \[
  d < 20
  \]

### 3. \( 18 \leq -3x \)
- Divide both sides by \(-3\) (again, reverse the inequality):
  \[
  \frac{18}{-3} \geq x
  \]
  \[
  -6 \geq x
  \]
  This can be rewritten as:
  \[
  x \leq -6
  \]

### 4. \( -5y \geq -35 \)
- Divide both sides by \(-5\):
  \[
  y \leq \frac{-35}{-5}
  \]
  \[
  y \leq 7
  \]

### 5. \( -4w - 13 > -21 \)
- Add 13 to both sides:
  \[
  -4w > -21 + 13
  \]
  \[
  -4w > -8
  \]
- Divide by \(-4\):
  \[
  w < \frac{-8}{-4}
  \]
  \[
  w < 2
  \]

### 6. \( 8z - 9 \geq -15 \)
- Add 9 to both sides:
  \[
  8z \geq -15 + 9
  \]
  \[
  8z \geq -6
  \]
- Divide by 8:
  \[
  z \geq \frac{-6}{8}
  \]
  \[
  z \geq -\frac{3}{4}
  \]

### 7. \( s \geq \frac{s + 6}{5} \)
- Multiply both sides by 5 to eliminate the fraction:
  \[
  5s \geq s + 6
  \]
- Subtract \(s\) from both sides:
  \[
  4s \geq 6
  \]
- Divide by 4:
  \[
  s \geq \frac{6}{4}
  \]
  \[
  s \geq \frac{3}{2}
  \]

### 8. \( \frac{2x - 9}{4} \leq x + 2 \)
- Multiply both sides by 4:
  \[
  2x - 9 \leq 4(x + 2)
  \]
  \[
  2x - 9 \leq 4x + 8
  \]
- Subtract \(2x\) from both sides:
  \[
  -9 \leq 2x + 8
  \]
- Subtract 8 from both sides:
  \[
  -9 - 8 \leq 2x
  \]
  \[
  -17 \leq 2x
  \]
- Divide by 2:
  \[
  \frac{-17}{2} \leq x
  \]
  This can be rewritten as:
  \[
  x \geq -\frac{17}{2}
  \]

Would you like me to provide the graphs for these inequalities? Let me know if you have any specific questions!

---

Here are 5 related questions you could explore next:

1. How do we graph the solution sets of inequalities on a number line?
2. What happens when we multiply or divide both sides of an inequality by a negative number?
3. Can inequalities have infinite solutions? Why or why not?
4. How do compound inequalities (such as \(a < x \leq b\)) work?
5. How do inequalities apply in real-world scenarios?

### Tip:
Always remember to reverse the inequality sign when multiplying or dividing both sides by a negative number!

Solve each inequality. Then graph the solution set on a number line.

Solve and graph a set of inequalities, covering algebraic concepts such as adding, subtracting, multiplying, and dividing inequalities. Learn how to handle inequalities when multiplying or dividing by negative numbers. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation