Solve the Absolute Value Inequality for y: |y-3|≤15

The given absolute value inequality is:

\[
|y - 3| \leq 15
\]

### Step 1: Rewrite the inequality without absolute value

The general rule for an absolute value inequality like \( |x| \leq a \) is:

\[
-a \leq x \leq a
\]

So, applying this rule to \( |y - 3| \leq 15 \):

\[
-15 \leq y - 3 \leq 15
\]

### Step 2: Solve for \( y \)

To isolate \( y \), add 3 to all parts of the inequality:

\[
-15 + 3 \leq y \leq 15 + 3
\]

\[
-12 \leq y \leq 18
\]

### Final Solution:

\[
y \in [-12, 18]
\]

This means that \( y \) can be any value between \( -12 \) and \( 18 \), inclusive.

Would you like more details on how absolute value inequalities work?

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Here are 5 related questions:
1. How do you solve \( |y - 5| \leq 10 \)?
2. What happens when the inequality is \( |y - 3| > 15 \)?
3. Can absolute value inequalities have no solutions?
4. How do you graph the solution \( y \in [-12, 18] \)?
5. What if the inequality was \( |y - 3| \geq 15 \)?

**Tip:** Absolute value inequalities often require splitting into two inequalities, one positive and one negative.

Learn how to solve the absolute value inequality |y - 3| ≤ 15 with a detailed step-by-step explanation. This solution covers core algebraic concepts, such as absolute value and inequalities, and provides an easy approach suitable for students in Grades 7-9.

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