# How do you solve 4x<2x-2<10?

Mar 22, 2018

The solution is $x < - 1$.

#### Explanation:

You can split up the compound inequality into two, smaller inequalities, like this:

$\textcolor{w h i t e}{\implies} 4 x < 2 x - 2 < 10$

$\implies \left(4 x < 2 x - 2\right) \mathmr{and} \left(2 x - 2 < 10\right)$

We will get one solution from each of the inequalities, and we have to find where they "overlap," or in other words, where their values are the same (this will make more sense later).

Solving each inequality:

$\textcolor{w h i t e}{\implies} \left(4 x < 2 x - 2\right) \mathmr{and} \left(2 x - 2 < 10\right)$

$\textcolor{w h i t e}{\implies 4 x} \left(2 x < - 2\right) \mathmr{and} \left(2 x < 12\right)$

$\textcolor{w h i t e}{\implies 4 2 x} \left(x < - 1\right) \mathmr{and} \left(x < 6\right)$

Now, if we graph these points on a number line, it will be easier to see where their lines overlap:

The inequality $x < - 1$ is shown in red, and $x < 6$ is shown in blue:

We can see that the only place where both lines are in common is where $x < - 1$. Therefore, that is the solution to our inequality.

Hope this helped!