# How do you solve and write the following in interval notation: 2x - 4< 4 or x + 5 ≤ 2 - 2x?

Oct 20, 2017
1. $x < 4$ and in interval notation $x \in \left(- \infty , 4\right)$
2. $x \le - 1$ and in interval notation $x \in \left(- \infty , - 1\right]$

#### Explanation:

Lets take the first example.

$2 x - 4 < 4$

This is an inequality. Inequalities are usually solved like regular equations.

Lets add $4$ to both sides of the equation.

$2 x - 4 + \textcolor{red}{4} < 4 + \textcolor{red}{4}$

$2 x < \textcolor{red}{8}$

Divide both sides by $2$

$\frac{2 x}{\textcolor{red}{2}} < \frac{8}{\textcolor{red}{2}}$

$x < 4$

This inequality means that the value of $x$ has to be less than $4$ to satisfy the inequality.

In interval notation it will be written as $x \in \left(- \infty , 4\right)$ because the value of $x$ can be any number between $- \infty$ and $4$ but cannot be $- \infty$ or $4$ hence we use this ( ) bracket.

Similarly when we simplify $x + 5 \le 2 - 2 x$ we get $x \le - 1$

So here the interval notation is $x \in \left(- \infty , - 1\right]$ because $x$ can be any number between $- \infty$ and $- 1$ and can be $- 1$ but cannot be $- \infty$ hence this ( ] bracket.