We are given the equation:

\[
|x + 7| + 2 = 4x
\]

To solve this, we need to handle the absolute value. The equation splits into two cases based on the definition of absolute value:

### Case 1: \(x + 7 \geq 0\) (i.e., \(x \geq -7\))
In this case, \(|x + 7| = x + 7\), so the equation becomes:

\[
x + 7 + 2 = 4x
\]

Simplifying:

\[
x + 9 = 4x
\]

Subtract \(x\) from both sides:

\[
9 = 3x
\]

Now, divide by 3:

\[
x = 3
\]

Since \(x = 3 \geq -7\), this solution is valid for Case 1.

### Case 2: \(x + 7 < 0\) (i.e., \(x < -7\))
In this case, \(|x + 7| = -(x + 7)\), so the equation becomes:

\[
-(x + 7) + 2 = 4x
\]

Simplifying:

\[
-x - 7 + 2 = 4x
\]

\[
-x - 5 = 4x
\]

Add \(x\) to both sides:

\[
-5 = 5x
\]

Now, divide by 5:

\[
x = -1
\]

However, this solution does not satisfy the condition \(x < -7\), so it is not valid for Case 2.

### Final Solution:
The only valid solution is \(x = 3\).

---

Would you like more details or further clarification?

Here are 5 related questions you could explore:
1. What happens when absolute value equations have no solution?
2. How do we solve absolute value inequalities?
3. What if the equation had a quadratic term?
4. How do we graph absolute value functions?
5. Can we solve systems involving absolute values?

**Tip:** When solving absolute value equations, always check if your solutions fit the conditions of each case!

This page explains how to solve the absolute value equation |x + 7| + 2 = 4x by breaking it into two cases: one for x >= -7 and another for x < -7. Learn step-by-step how to solve the equation and find that the only valid solution is x = 3. Suitable for students in Grades 9-10.

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