https://file.aiquickdraw.com/mathbot/file/1726404629160jhz55bf1.jpg

Let's solve each part of the problem step by step:

### a) \(\left|\frac{x - 2}{3}\right| = 5\)

The absolute value equation can be broken into two cases:

1. \(\frac{x - 2}{3} = 5\)
   
   Solving for \(x\):
   \[
   x - 2 = 15 \quad (\text{Multiply both sides by 3})
   \]
   \[
   x = 17
   \]

2. \(\frac{x - 2}{3} = -5\)
   
   Solving for \(x\):
   \[
   x - 2 = -15 \quad (\text{Multiply both sides by 3})
   \]
   \[
   x = -13
   \]

**Solution for (a):**
\[
x = 17 \quad \text{or} \quad x = -13
\]

### b) \(\left|\frac{2x - 1}{3}\right| \leq 1\)

For absolute value inequalities, we rewrite this as:
\[
-1 \leq \frac{2x - 1}{3} \leq 1
\]

Now, solve the inequality by multiplying all parts by 3:
\[
-3 \leq 2x - 1 \leq 3
\]

Add 1 to all parts:
\[
-2 \leq 2x \leq 4
\]

Finally, divide by 2:
\[
-1 \leq x \leq 2
\]

**Solution for (b):**
\[
x \in [-1, 2]
\]

### c) \(\left|4x - 1\right| > 5\)

This absolute value inequality can be split into two cases:

1. \(4x - 1 > 5\)

   Solving for \(x\):
   \[
   4x > 6 \quad (\text{Add 1 to both sides})
   \]
   \[
   x > \frac{6}{4} = 1.5
   \]

2. \(4x - 1 < -5\)

   Solving for \(x\):
   \[
   4x < -4 \quad (\text{Add 1 to both sides})
   \]
   \[
   x < -1
   \]

**Solution for (c):**
\[
x < -1 \quad \text{or} \quad x > 1.5
\]

This would be represented as the union of intervals:
\[
x \in (-\infty, -1) \cup (1.5, \infty)
\]

---

Would you like me to plot these solutions on number lines for visual clarity? Let me know if you need further details!

### Follow-up Questions:
1. How are absolute value inequalities solved differently from absolute value equations?
2. What is the difference between "or" and "and" when dealing with inequalities?
3. How do we graph compound inequalities on a number line?
4. Can you explain why multiplying or dividing by a negative number flips the inequality sign?
5. What other types of problems use absolute value inequalities?

### Tip:
When solving absolute value inequalities, always consider whether the inequality sign indicates an "and" (intersection) or "or" (union) condition for the solutions.

Solve the following inequalities and graph the values in the number line including the interval: a) |(x - 2)/3| = 5, b) |(2x - 1)/3| ≤ 1, c) |4x - 1| > 5

Learn how to solve absolute value equations and inequalities with step-by-step solutions. Understand concepts such as breaking absolute value into cases, solving linear inequalities, and graphing solutions on the number line. Suitable for students in Grades 8-10.

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