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 $x = 3$.

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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!