Solve Absolute Value Equation: 5x + 2|x + 2| = 10

Published on July 6, 2026

Algebra

Learn how to solve linear equations involving absolute values by considering different cases for the expression inside the absolute value.

Question

teach me how to solve this

Answer

To solve the equation

, we must account for the absolute value term. The expression inside the absolute value,

, changes its behavior depending on whether

is non-negative or negative.

We split the problem into two cases based on the critical point

Case 1: (which means )

In this case, the absolute value

is simply

. We substitute this into the equation:

Now, solve for

Distribute the :
Combine like terms:
Subtract from both sides:
Divide by :

Check validity: Since

, it satisfies the condition

. Therefore,

is a valid solution.

Case 2: (which means )

In this case, the absolute value

becomes

. We substitute this into the equation:

Now, solve for

Distribute the :
Combine like terms:
Add to both sides:
Divide by :

Check validity:

. However, our condition for this case was

. Since

is not less than

, this solution is extraneous and must be rejected.

Final Answer:

The only valid solution is:

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