|x|+x=5

To solve the equation \(|x| + x = 5\), we need to consider two cases because of the absolute value.

Case 1: \(x \geq 0\)
In this case, \(|x| = x\). Substituting into the equation, we get: \[ x + x = 5 \implies 2x = 5 \implies x = \frac{5}{2} \] Since \(\frac{5}{2} \geq 0\), this solution is valid.

Case 2: \(x < 0\)
In this case, \(|x| = -x\). Substituting into the equation, we get: \[ -x + x = 5 \implies 0 = 5 \] This is a contradiction, so there are no valid solutions in this case.

The only solution to the equation \(|x| + x = 5\) is: \[ x = \frac{5}{2} \]