Question

Question
Express all real numbers less than -2 or greater than or equal to 3 in interval notation.

Answer 1

Real numbers can be expressed using the following interval,

$\mathbb{R}=(-\infty,\infty)$

Of course infinities are not just normal infinities but thats out of the scope of this question.

Real numbers less than two can be expressed with,

$(-\infty,\infty)\cap(-\infty,-2)=\boxed{(-\infty,-2)}$

The $\cap$ is called intersection ie. where are both intervals valid. First we took real numbers then we intersected them with real numbers valued less than -2 and we got real numbers which are less than -2.

Similarly we can perform with "greater than or equal to 3" real numbers,

$(-\infty,\infty)\cap[3,\infty)=\boxed{[3,\infty)}$

So we have one interval stretching from negative infinity to (but not including) -2, and another interval stretching from including 3 to positive infinity.

If we want numbers in both intervals we can express this two ways,

First way is to use $\cup$ union operator to denote we want numbers from two intervals,

$\boxed{(-\infty,2)\cup[3,\infty)}$

The second way is to specify which numbers we do not want, we do not want -2 and everything up to but not including 3, which is expressed with the following interval

$[-2,3)$

Now we just take out the not wanted interval from real numbers and we will remain with all wanted numbers,

$\boxed{(-\infty,\infty)-[-2,3)}$

Hope this helps.