# What is the solution set for abs(x – 2) + 4 = 3?

Aug 26, 2015

$x = \emptyset$

#### Explanation:

As it is written, your absolute value equation has no real solutions.

That happens because the absolute value of any real number always returns the positive value of said number, irrespective of its sign.

In your case, you know that if you isolate the modulus on one side of the equation by adding $- 4$ to both sides, you get

$| x - 2 | + \textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} - \textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} = 3 - 4$

$| x - 2 | = - 1$

In this case, you need the absolute value of a real number, which is what $x - 2$ is if $x \in \mathbb{R}$, to be negative.

This contradicts the definition of the absolute value, which implies that $x \notin \mathbb{R}$. Therefore, the equation has no real solutions.