# How do you solve abs(5-x)<=x?

Oct 4, 2017

Use the piecewise definition of the absolute value function, |f(x)|={(f(x);f(x)>=0),(-f(x);f(x)<0):} to separate into two inequalities.
Solve both inequalities.

#### Explanation:

Separate into two inequalities:

$5 - x \le x$ and $- \left(5 - x\right) \le x$

$5 - x \le x$ and $- 5 + x \le x$

Add x to both sides of the first inequality and subtract x from both sides of the second:

$5 \le 2 x$ and $- 5 \le 0$

The second inequality does not depend on x, therefore, we discard it.

Divide both sides of the first inequality by 2:

$\frac{5}{2} \le x$

Flip the inequality

$x \ge \frac{5}{2}$