https://file.aiquickdraw.com/mathbot/file/1727187564007s68f9fzo.jpg

We are given the equation:

\[
4 |2w + 7| - 49 = -5
\]

Let's solve for \( w \) step by step.

### Step 1: Isolate the absolute value term
First, move the \(-49\) to the other side of the equation:

\[
4 |2w + 7| = -5 + 49
\]

\[
4 |2w + 7| = 44
\]

### Step 2: Divide by 4

\[
|2w + 7| = \frac{44}{4}
\]

\[
|2w + 7| = 11
\]

### Step 3: Solve the absolute value equation
We now have:

\[
|2w + 7| = 11
\]

This implies two possible cases:
1. \( 2w + 7 = 11 \)
2. \( 2w + 7 = -11 \)

#### Case 1: \( 2w + 7 = 11 \)

Solve for \( w \):

\[
2w = 11 - 7
\]

\[
2w = 4
\]

\[
w = 2
\]

#### Case 2: \( 2w + 7 = -11 \)

Solve for \( w \):

\[
2w = -11 - 7
\]

\[
2w = -18
\]

\[
w = -9
\]

### Final Answer:
The two solutions for \( w \) are:

\[
w = 2, -9
\]

Do you have any questions or need further clarification on any of the steps?

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Here are 5 related questions to explore further:
1. How would the solution change if the equation were \( 4 |2w + 7| - 49 = 5 \)?
2. What are the properties of absolute value equations?
3. How would you graph the absolute value function \( |2w + 7| \)?
4. Can absolute value equations ever have no solution? If so, when?
5. What happens if the absolute value equation involves quadratic terms?

**Tip:** When solving absolute value equations, always remember to consider both the positive and negative cases!

Learn how to solve the absolute value equation 4 |2w + 7| - 49 = -5 with a detailed, step-by-step solution. This problem covers key algebraic concepts like absolute value and linear equations. Suitable for students in Grades 7-9.

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