Question

How do you write the following in interval notation: \[x > 2\] OR $5 > x$?

Answer 1

Hint: We start solving the problem by finding the interval of numbers that were included in \[x > 2\]. We then represent all these numbers in the interval on a number line. We then find the interval of numbers that were included in $5 > x$. We then represent all these obtained numbers in the interval on a number line. We then represent all the numbers in both the intervals on the number line and find the union of both the intervals to get the required answer.

Complete step by step answer:
According to the problem, we are asked to write the given expression \[x > 2\] OR $5 > x$ in interval notation.
Let us first represent each of the given intervals on a number line.
We know that $x > 2$ represents all the real numbers that were greater than 2 i.e., $\left( 2,\infty \right)$ ---(1). Let us represent this on the number line.

We know that $5 > x$ represents all the real numbers that were less than 5 i.e., $\left( -\infty ,5 \right)$ ---(2). Let us represent this on the number line.

Now, we need to find the values that were included in the expression: \[x > 2\] OR $5 > x$.
We know that OR is represented by union $\left( \cup \right)$. So, we get $\left( x > 2 \right)\cup \left( 5 > x \right)$.
From equations (1) and (2), we get $\left( x > 2 \right)\cup \left( 5 > x \right)=\left( 2,\infty \right)\cup \left( -\infty ,5 \right)$, which means that we need to include all the numbers present in both intervals. Let us represent this interval on the number line.

From the figure, we can see that the given interval represents all the real numbers on the number line. So, we can represent it with R or $\left( -\infty ,\infty \right)$.
$\therefore $ We have found the interval notation for the given expression \[x > 2\] OR $5 > x$ as $\left( -\infty ,\infty \right)$.

Note:
We should not confuse \[x > 2\] OR $5 > x$ with \[x > 2\] AND $5 > x$ which is the common mistake done by students. We should take the values of x in which the given expression \[x > 2\] OR $5 > x$ is true. Whenever we get this type of problem, we first represent the intervals on a number line to proceed to the solution. Similarly, we can expect problems to write the following in interval notation: \[x > 2\] AND $x > 5$.