Question

Express the inequality in interval notation, and then graph the corresponding interval.

$x\le 1$

Short answer

The inequality form of the given interval $\left(-\infty ,1\right]$.

Step-by-step solution

Step 1. Apply the concept of interval and inequality.

If $a<b$, then the open interval from a to b consists of all numbers between a and b and is denoted by $\left(a,b\right)$. The closed interval from a to b includes the endpoints and is denoted by $\left[a,b\right]$.

Using the set-builder form, we can write the intervals as inequality:

localid="1645531218422" $$\begin{gathered} \left(a,b\right)=\left\{x|a<x<b\right\} \\ \left[a,b\right]=\left\{x|a\le x\le b\right\} \end{gathered}$$

Step 2. Express in terms of inequality.

Given inequality is$x\le 1$

This includes all the numbers less than and equal to 1.

Using set-builder form to express the inequality:

$\left\{x\right|x\le 1\}$

So, the interval is given by$\left(-\infty ,1\right]$

Step 3. Graph the interval.

From step 2, we have $\left(-\infty ,1\right]$

This represents all the numbers less than and 1.

Since 1 is included in the interval, it will be represented by a closed circle.

So, the graph will be: