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We are given the inequality:

\[
y + 3 < |2x| - 5
\]

### Step 1: Isolate \( y \)

First, subtract 3 from both sides:

\[
y < |2x| - 5 - 3
\]

This simplifies to:

\[
y < |2x| - 8
\]

### Step 2: Break down the absolute value

To solve the inequality involving the absolute value, we must handle two cases: one for when \( 2x \geq 0 \) (which implies \( x \geq 0 \)), and one for when \( 2x < 0 \) (which implies \( x < 0 \)).

#### Case 1: \( x \geq 0 \)

For \( x \geq 0 \), \( |2x| = 2x \), so the inequality becomes:

\[
y < 2x - 8
\]

#### Case 2: \( x < 0 \)

For \( x < 0 \), \( |2x| = -2x \), so the inequality becomes:

\[
y < -2x - 8
\]

### Step 3: Compare to the given answer choices

We now have the following cases:

- For \( x \geq 0 \), \( y < 2x - 8 \)
- For \( x < 0 \), \( y < -2x - 8 \)

These match answer **b**, where:
- \( y < -2x - 8 \) for \( x < 0 \)
- \( y < 2x - 8 \) for \( x \geq 0 \)

Thus, the correct solution is **b**.

Would you like any further details or clarifications?

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Here are five related questions for further practice:

1. How do you solve inequalities involving absolute values?
2. What are the different methods to solve piecewise functions?
3. How does the behavior of the inequality change when \( |x| \) is involved?
4. How would the inequality change if \( y + 3 \leq |2x| - 5 \) were given?
5. Can you graph this piecewise inequality and explain its regions?

**Tip:** When solving absolute value inequalities, always break it into two cases (positive and negative) to handle the different behaviors of the absolute value function.

What is the solution of the inequality y + 3 < |2x| - 5?

Learn how to solve the inequality y + 3 < |2x| - 5 by breaking down the absolute value function into two cases. This step-by-step solution helps understand inequalities involving absolute values and is suitable for high school students.

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