# How do you solve 4abs[x+1]-2<10?

Nov 2, 2017

See a solution process below:

#### Explanation:

First, add $\textcolor{red}{2}$ to each side of the inequality to isolate the absolute value term while keeping the inequality balanced:

$4 \left\mid x + 1 \right\mid - 2 + \textcolor{red}{2} < 10 + \textcolor{red}{2}$

$4 \left\mid x + 1 \right\mid - 0 < 12$

$4 \left\mid x + 1 \right\mid < 12$

Now, divide each side of the inequality by $\textcolor{red}{4}$ to isolate the absolute value function while keeping the inequality balanced:

$\frac{4 \left\mid x + 1 \right\mid}{\textcolor{red}{4}} < \frac{12}{\textcolor{red}{4}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} \left\mid x + 1 \right\mid}{\cancel{\textcolor{red}{4}}} < 3$

$\left\mid x + 1 \right\mid < 3$

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

$- 3 < x + 1 < 3$

Now, subtract $\textcolor{red}{1}$ from each segment of the system of inequalities to solve for $x$ while keeping the system balanced:

$- 3 - \textcolor{red}{1} < x + 1 - \textcolor{red}{1} < 3 - \textcolor{red}{1}$

$- 4 < x + 0 < 2$

$- 4 < x < 2$

Or

$x > - 4$ and $x < 2$

Or, in interval notation:

$\left(- 4 , 2\right)$