https://file.aiquickdraw.com/mathbot/file/17268879986094t8h9bxf.jpg

Let's solve each of the absolute value inequalities step by step.

### a) \( |3x| > 21 \)
- Start by removing the absolute value:
  \[
  3x > 21 \quad \text{or} \quad 3x < -21
  \]
- Solving for \( x \):
  \[
  x > 7 \quad \text{or} \quad x < -7
  \]
- The solution in interval notation is:
  \[
  (-\infty, -7) \cup (7, \infty)
  \]

### b) \( |14x - 5| < 51 \)
- Remove the absolute value:
  \[
  -51 < 14x - 5 < 51
  \]
- Solve for \( x \) by first adding 5 to all parts:
  \[
  -46 < 14x < 56
  \]
- Divide by 14:
  \[
  -\frac{46}{14} < x < \frac{56}{14}
  \]
- Simplify:
  \[
  -\frac{23}{7} < x < 4
  \]
- The solution in interval notation is:
  \[
  \left( -\frac{23}{7}, 4 \right)
  \]

### c) \( |3x + 3| \geq 9 \)
- Remove the absolute value:
  \[
  3x + 3 \geq 9 \quad \text{or} \quad 3x + 3 \leq -9
  \]
- Solve for \( x \) in both cases:
  \[
  3x \geq 6 \quad \Rightarrow \quad x \geq 2
  \]
  \[
  3x \leq -12 \quad \Rightarrow \quad x \leq -4
  \]
- The solution in interval notation is:
  \[
  (-\infty, -4] \cup [2, \infty)
  \]

### d) \( \left| \frac{3x - 6}{4} \right| \leq 17 \)
- Remove the absolute value:
  \[
  -17 \leq \frac{3x - 6}{4} \leq 17
  \]
- Multiply all parts by 4:
  \[
  -68 \leq 3x - 6 \leq 68
  \]
- Add 6 to all parts:
  \[
  -62 \leq 3x \leq 74
  \]
- Divide by 3:
  \[
  -\frac{62}{3} \leq x \leq \frac{74}{3}
  \]
- The solution in interval notation is:
  \[
  \left[ -\frac{62}{3}, \frac{74}{3} \right]
  \]

### Summary
- Interval notation answers:
  - a) \( (-\infty, -7) \cup (7, \infty) \)
  - b) \( \left( -\frac{23}{7}, 4 \right) \)
  - c) \( (-\infty, -4] \cup [2, \infty) \)
  - d) \( \left[ -\frac{62}{3}, \frac{74}{3} \right] \)

If you'd like to see any of these on a real number line or need more details, let me know!

### Relative Questions:
1. How do you solve absolute value equations involving inequalities?
2. What happens when you multiply or divide by a negative number in inequalities?
3. Can absolute value inequalities have no solution? If so, under what conditions?
4. How would these inequalities change if they involved a "less than" instead of a "greater than" sign?
5. How do you graph solutions of absolute value inequalities on a number line?

### Tip:
Always remember to flip the inequality sign when multiplying or dividing by a negative number in inequalities.

Solve the following Absolute Value Inequalities, providing two of your answers in Interval Notation and two of your answers on a Real Number Line:

a) |3x| > 21
b) |14x - 5| < 51
c) |3x + 3| ≥ 9
d) |(3x - 6)/4| ≤ 17

Learn how to solve absolute value inequalities including |3x| > 21 and |14x - 5| < 51 with detailed steps. This guide covers interval notation, graphing on a real number line, and key properties of inequalities, ideal for students in Grades 8-10.

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