# How do you solve and write the following in interval notation: x + 2 > -3 and x + 2 <5?

Dec 21, 2016

$x \in \left(\text{-} 5 , 3\right)$.

#### Explanation:

Each one of these two inequalities has a set of $x$-values that make it true. Solving the two inequalities together means finding the $x$-values that are in both sets.

From the first inequality, we have

$x + 2 > \text{-"3 => x+2- color(blue)(2)>"-3} - \textcolor{b l u e}{2}$

color(white)(x+2> "-"3) => x"               ">"-5"

Alright; so the first inequality is true for every $x$ greater than -5:
$x \in \left(\text{-5} , \infty\right)$

Similarly, we can solve the second inequality as follows:

$x + 2 < 5 \implies x + 2 - \textcolor{b l u e}{2} < 5 - \textcolor{b l u e}{2}$

$\textcolor{w h i t e}{x + 2 < 5} \implies x \text{ } < 3$

Okay, so the second inequality is true for every $x$ less than 3:
$x \in \left(\text{-} \infty , 3\right)$

We then ask ourselves: where do these two sets overlap? In other words, what values of $x$ make both inequalities true? We can graph both sets on a number line to help us see this:

$\text{ o————————————> "x>"-5}$
$\text{<————————————o } x < 3$
<————— -5 ————— 0 ——— 3 —————>

From the number line above, we see that the two solutions overlap between $x = \text{-5}$ and $x = 3$ (not including the endpoints). In interval notation, this is written by combining the two individual solution intervals:

$x \in \left(\text{-5", oo) nn ("-} \infty , 3\right)$

which, as we can see from the number line, simplifies to

$x \in \left(\text{-} 5 , 3\right)$.