# How do you solve and write the following in interval notation: x − 3 ≤ 2 AND -x / 3 &lt; 2?

Jan 21, 2017

$x - 3 \le 2$ and $- \frac{x}{3} < 2$ $\iff$ $x \in \left(- 6 , 5\right]$

#### Explanation:

$x - 3 \le 2$ and $- \frac{x}{3} < 2$

Then

for $x - 3 \le 2$

$\iff$

$x \le 2 + 3 = 5$ Add three to both sides

Therefore

$x \le 5 \iff x \in \left(- \infty , 5\right]$

This means $x$ is in the set of points between $- \infty$ to and including $5$. So, $x$ could be $5$ or any number less than $5$.

likewise

for $- \frac{x}{3} < 2$

$- x < 2 \left(3\right) = 6$ multiply both sides by 3

Then

$- x < 6$ divide both sides by -1

$\iff x > - 6$ then $x \in \left(- 6 , \infty\right)$

This means $x$ is in the set of points greater than $- 6$ but not including $- 6$. Then $x$ can be any number greater than $- 6$ but not $- 6$.

If $x \le 5$ and $x > - 6$, then $x \in \left(- \infty , 5\right] \cap \left(- 6 , \infty\right)$. Where the upside down cup $\cap$ means "intersection", meaning $x$ is in both the first and second set. Then $x \in \left(- 6 , 5\right]$ because $x$ cannot be less than or equal $- 6$, and it is also less than any number greater than $5$.