# How do you solve and graph 2x < 10 and -5x <5?

Oct 23, 2017

See a solution process below:

#### Explanation:

To solve the first inequality, divide each side of the inequality by $\textcolor{red}{2}$ to solve for $x$ while keeping the inequality balanced:

$\frac{2 x}{\textcolor{red}{2}} < \frac{10}{\textcolor{red}{2}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} x}{\cancel{\textcolor{red}{2}}} < 5$

$x < 5$

To solve the second inequality, divide each side of the inequality by $\textcolor{b l u e}{- 5}$ to solve for $x$ while keeping the inequality balanced. However, because we are multiplying or dividing an inequality by a negative term we must reverse the inequality operator:

$\frac{- 5 x}{\textcolor{b l u e}{- 5}} \textcolor{red}{>} \frac{5}{\textcolor{b l u e}{- 5}}$

$\frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 5}}} x}{\cancel{\textcolor{b l u e}{- 5}}} \textcolor{red}{>} - 1$

$x > - 1$

The solution is: $x > - 1$ and $x < 5$

Or, in interval notation: $\left(- 1 , 5\right)$

To graph this we will draw two vertical lines at $- 1$ and $5$ on the horizontal axis.

The lines will be dashed lines because the inequality operators do not contain an "or equal to" clause.

We will shade between the interval of the lines: