Question

Solve the equation. $$|x-5|=2$$

Short answer

The solutions for the given equation are \(x = 3\) and \(x = 7\).

Step-by-step solution

Set Up Two Equations

Since the absolute value of an equation represents a distance, and distances can be both to the left and right on a number line, we consider two different possibilities for the value of x: one where x - 5 equals 2 and another where x - 5 equals -2. The two equations to solve are therefore \(x - 5 = 2\) and \(x - 5 = -2\).

Solve First Equation

In the first equation, \(x - 5 = 2\), add 5 to both sides to get \(x = 2 + 5 = 7\). So, one possible solution for x is 7.

Solve Second Equation

In the second equation, \(x - 5 = -2\), add 5 to both sides to get \(x = -2 + 5 = 3\). So, another possible solution for x is 3. Hence, \(x\) can be 3 or 7 in the given equation \(|x - 5| = 2.\)