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.