Write the answer as a solution set, using interval notation, and as a number line graph.     3|5x – 4| + 3 ≤ 15  

Let's solve the inequality step by step and write the solution in interval notation and represent it on a number line.

### Problem:
\[
3|5x - 4| + 3 \leq 15
\]

### Step 1: Isolate the absolute value expression
Subtract 3 from both sides:
\[
3|5x - 4| \leq 12
\]

### Step 2: Divide both sides by 3
\[
|5x - 4| \leq 4
\]

### Step 3: Solve the absolute value inequality
The inequality \(|A| \leq B\) means \(-B \leq A \leq B\). Therefore, we can write:
\[
-4 \leq 5x - 4 \leq 4
\]

### Step 4: Solve for \(x\)
Add 4 to all parts of the inequality:
\[
0 \leq 5x \leq 8
\]

Now, divide by 5:
\[
0 \leq x \leq \frac{8}{5}
\]

### Step 5: Write the solution in interval notation
The solution in interval notation is:
\[
[0, \frac{8}{5}]
\]

### Step 6: Draw the number line graph
- Draw a number line.
- Mark points 0 and \(\frac{8}{5}\).
- Use closed circles at both 0 and \(\frac{8}{5}\) because the inequality includes the endpoints.
- Shade the region between 0 and \(\frac{8}{5}\), indicating all the values of \(x\) that satisfy the inequality.

This is the solution set and its graph representation.

---

Do you want further details, or would you like me to generate the number line graph for you?

Here are 5 related questions to explore:
1. How would the solution change if the inequality were strict (\(<\) instead of \(\leq\))?
2. What happens if you change the constant 15 to another number?
3. Can you explain how the solution set would be affected if the absolute value expression involved subtraction instead?
4. How would the graph look if the solution was unbounded?
5. What if you had multiple absolute value terms in the inequality?

**Tip:** Always isolate the absolute value expression before solving inequalities involving absolute values.

Write the answer as a solution set, using interval notation, and as a number line graph. 3|5x – 4| + 3 ≤ 15

Step-by-step solution to solve the absolute value inequality 3|5x - 4| + 3 ≤ 15. Learn how to isolate the absolute value expression, solve the inequality, and write the solution in interval notation. Includes graph representation on a number line. Suitable for Grades 9-11.

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