Question

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

Answer 1

`x in ("-"5, 3)`.

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> "-"3 => x+2- color(blue)(2)>"-3" - color(blue)(2)`

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

Alright; so the first inequality is true for every `x` greater than -5:
`x in ("-5", oo)`

Similarly, we can solve the second inequality as follows:

`x+2<5 => x+2-color(blue)2 < 5 - color(blue)2`

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

Okay, so the second inequality is true for every `x` less than 3:
`x in ("-"oo, 3)`

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:

`"                   o————————————>       "x>"-5"`
`"<————————————o                          "x<3`
<————— -5 ————— 0 ——— 3 —————>

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

`x in ("-5", oo) nn ("-"oo, 3)`

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

`x in ("-"5, 3)`.