solve the inequality. Write your answer using interval notation.  |3x − 5| ≤ 4

To solve the inequality \( |3x - 5| \leq 4 \), we need to break it down into two cases because the absolute value expression \( |A| \leq B \) can be rewritten as two simultaneous inequalities:

\[
-4 \leq 3x - 5 \leq 4
\]

### Step 1: Solve the two inequalities
We will now split the inequality into two parts and solve for \( x \).

#### First part: \( 3x - 5 \geq -4 \)
1. Add 5 to both sides:
   \[
   3x \geq -4 + 5
   \]
   \[
   3x \geq 1
   \]
2. Divide both sides by 3:
   \[
   x \geq \frac{1}{3}
   \]

#### Second part: \( 3x - 5 \leq 4 \)
1. Add 5 to both sides:
   \[
   3x \leq 4 + 5
   \]
   \[
   3x \leq 9
   \]
2. Divide both sides by 3:
   \[
   x \leq 3
   \]

### Step 2: Combine the two parts
Now, we combine the two results:
\[
\frac{1}{3} \leq x \leq 3
\]

### Step 3: Express the solution in interval notation
The solution is the set of \( x \)-values that satisfy both inequalities. In interval notation, this is:
\[
\left[ \frac{1}{3}, 3 \right]
\]

Thus, the solution to the inequality \( |3x - 5| \leq 4 \) is \( \left[ \frac{1}{3}, 3 \right] \).

Would you like any additional details or have further questions?

Here are five related questions:
1. How do you solve an inequality involving a strict inequality \( |A| < B \)?
2. What changes when solving inequalities with \( |A| \geq B \)?
3. Can you solve \( |3x - 5| \geq 4 \) and express the answer in interval notation?
4. How do you solve absolute value equations, e.g., \( |3x - 5| = 4 \)?
5. What happens when an absolute value inequality has no solution?

**Tip:** When solving absolute value inequalities, always consider breaking it into two parts, one with a positive and one with a negative bound!

solve the inequality. Write your answer using interval notation. |3x − 5| ≤ 4

Learn how to solve the absolute value inequality |3x − 5| ≤ 4 with step-by-step explanations. This problem covers essential algebraic concepts, including breaking down inequalities into two parts, solving linear inequalities, and expressing solutions in interval notation. Suitable for students in Grades 9-11.

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