We are given the equation:

\[
|4 + x| - 5 = 7
\]

To solve for \( x \), let's follow these steps:

### Step 1: Isolate the absolute value expression
Add 5 to both sides of the equation to isolate the absolute value term:

\[
|4 + x| = 12
\]

### Step 2: Solve the absolute value equation
The absolute value equation \( |4 + x| = 12 \) means that \( 4 + x \) can be either 12 or -12. This gives us two cases to solve for \( x \):

1. \( 4 + x = 12 \)
2. \( 4 + x = -12 \)

### Step 3: Solve each case

**Case 1:**
\[
4 + x = 12
\]
Subtract 4 from both sides:
\[
x = 8
\]

**Case 2:**
\[
4 + x = -12
\]
Subtract 4 from both sides:
\[
x = -16
\]

### Step 4: Final Solution
The two solutions are:
\[
x = 8 \quad \text{or} \quad x = -16
\]

Do you need further details, or do you have any questions?

Here are 5 related questions for further practice:
1. Solve: \( |3 - 2x| - 4 = 6 \).
2. Solve: \( |x - 5| + 2 = 9 \).
3. Solve: \( |2x + 7| = 9 \).
4. What is the solution for \( |x + 3| + 1 = 10 \)?
5. Solve \( |5x - 1| = 14 \).

**Tip**: When solving absolute value equations, always split the problem into two cases: one for the positive value and one for the negative value.

Learn how to solve the absolute value equation ∣4 + x∣ − 5 = 7 with a step-by-step breakdown. Understand key algebraic concepts like isolating absolute value expressions and solving two cases for x. This problem is ideal for students in Grades 8-10.

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