We have the equation:

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

### Step 1: Isolate the absolute value expression
Subtract 2 from both sides:

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

### Step 2: Consider both cases for the absolute value

#### Case 1: \( x + 7 \geq 0 \) 
In this case, the absolute value function can be removed without changing the expression:

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

Solve for \(x\):

\[
7 + 2 = 4x - x
\]
\[
9 = 3x
\]
\[
x = 3
\]

#### Case 2: \( x + 7 < 0 \)
In this case, the absolute value function changes the sign of the expression inside:

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

Simplify:

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

Solve for \(x\):

\[
-7 + 2 = 4x + x
\]
\[
-5 = 5x
\]
\[
x = -1
\]

### Step 3: Check the validity of the solutions
1. For \(x = 3\), check if \(x + 7 \geq 0\):

\[
3 + 7 = 10 \geq 0 \quad \text{(Valid)}
\]

2. For \(x = -1\), check if \(x + 7 < 0\):

\[
-1 + 7 = 6 \quad \text{(Invalid, since it's positive)}
\]

Thus, the only valid solution is:

\[
\boxed{x = 3}
\]

---

Would you like further clarification or details? Here are some related questions you might want to explore:

1. What are the properties of absolute value functions?
2. How do we handle equations involving inequalities?
3. How can you graph the solution to an absolute value equation?
4. What are the steps to check the validity of solutions in absolute value equations?
5. How would this problem change if the absolute value was on both sides?

**Tip:** Always check the conditions of the absolute value to avoid extraneous solutions!

Learn how to solve the absolute value equation |x + 7| + 2 = 4x. This step-by-step solution walks through isolating the absolute value expression, considering both positive and negative cases, and verifying the validity of the solutions. Suitable for students in Grades 9-10.

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