# What is the solution set for absx - 1 < 4?

Aug 14, 2015

$- 5 < x < 5$

#### Explanation:

To solve this absolute value inequality, first isolate the modulus on one side by adding $1$ to both sides of the inequality

$| x | - \textcolor{red}{\cancel{\textcolor{b l a c k}{1}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{1}}} < 4 + 1$

$| x | < 5$

Now, depending on the possible sign of $x$, you have two possiblities to account for

• $x > 0 \implies | x | = x$

This means that the inequality becomes

$x < 5$

• $x < 0 \implies | x | = - x$

This time, you have

$- x < 5 \implies x > - 5$

These two conditions will determine the solution set for the absolute value inequality. Since the inequality holds true for $x > - 5$, any value of $x$ that's smaller than that will be excluded.

LIkewise, since $x < 5$, any value of $x$ greater than $5$ will also be excluded. This means that the solution set to this inequality will be $- 5 < x < 5$, or $x \in \left(- 5 , 5\right)$.