Let's analyze and solve the matching problem step-by-step.

You have a set of inequalities that need to be matched with corresponding number lines. I'll first solve each inequality, then determine the number line that corresponds to each solution.

Equations and Inequalities:

1. $-1 + \frac{2}{3} x > -2$ or $6x - 1 < -19$:

   • For $-1 + \frac{2}{3} x > -2$: $\frac{2}{3} x > -1 + 1 \implies \frac{2}{3} x > -1 \implies x > -\frac{3}{2}$ (one inequality is $x > -1.5$)

   • For $6x - 1 < -19$: $6x < -18 \implies x < -3$ (the other inequality is $x < -3$).

   Therefore, the solution to this inequality is $x > -1.5$ or $x < -3$, which suggests two disjoint intervals on the number line. This matches number line D, showing two separate sections of solutions on either side.

2. $2x + 7 = 1$ or $3x + 2 = 1$:

   • For $2x + 7 = 1$: $2x = -6 \implies x = -3$
   • For $3x + 2 = 1$: $3x = -1 \implies x = -\frac{1}{3}$

   So the solutions are $x = -3$ and $x = -\frac{1}{3}$. This matches number line C, which has dots at $x = -3$ and $x = -\frac{1}{3}$ indicating these points.

3. $-|3 - 2x| = -9$:

   • Simplify: $|3 - 2x| = 9$ This means either $3 - 2x = 9$ or $3 - 2x = -9$.

     • For $3 - 2x = 9$: $-2x = 6 \implies x = -3$
     • For $3 - 2x = -9$: $-2x = -12 \implies x = 6$

   So the solution is $x = -3$ or $x = 6$. This matches number line A, with points at $x = -3$ and $x = 6$.

4. $-1 < 3x < 6$:

   • Divide the inequality into two parts: $-1 < 3x \implies x > -\frac{1}{3}, \quad 3x < 6 \implies x < 2$

   So the solution is $-\frac{1}{3} < x < 2$. This matches number line E, with an open interval between $-\frac{1}{3}$ and 2.

5. $\frac{1}{4} x - 2 \leq 1$:

   • Solve for $x$: $\frac{1}{4}x \leq 3 \implies x \leq 12$

   So the solution is $x \leq 12$. This matches number line B, which shows all values less than or equal to 12.

Final Matching:

• D: $-1 + \frac{2}{3} x > -2$ or $6x - 1 < -19$
• C: $2x + 7 = 1$ or $3x + 2 = 1$
• A: $-|3 - 2x| = -9$
• E: $-1 < 3x < 6$
• B: $\frac{1}{4} x - 2 \leq 1$

Let me know if you want more details on any step! Here are five related questions for further practice:

1. How do you solve compound inequalities involving "or" and "and" conditions?
2. How do you graph inequalities on a number line?
3. What are absolute value equations and how do they affect solution sets?
4. How do you handle equations with fractions?
5. How do you know when to reverse inequality signs?

Tip: When solving compound inequalities, treat each part separately before combining them according to the conditions (e.g., "or" means union, "and" means intersection).