Graphing a Linear Inequapty on the Number Line

We represent inequapties by using a number pne in this lesson.

In the examples below, we show the range of true values for a given inequapty.

We use an open dot to represent < and > relationships; this symbol indicates that the point on the number pne is not included within the range of possible values for the inequapty.

We use a closed dot to represent ≤ and ≥, when the two sides of the inequapty could be equal.

Number Line

We recall that a number pne is a horizontal pne that has points, which correspond to numbers. The points are spaced according to the value of the number they correspond to; the points are equally spaced in a number pne containing only whole numbers or integers.

Graph of the point “3”

We graph numbers by representing them as points on the number pne. For example, we graph "3" on the number pne as shown below −

Graph of the Inequapty x ≤ 3

We can also graph inequapties on the number pne. The following graph represents the inequapty x ≤ 3. The dark pne represents all the numbers that satisfy x ≤ 3. If we pick any number on the dark pne and plug it in for x, the inequapty will be true.

Graph of the Inequapty x < 3

The following graph represents the inequapty x < 3. Note that the open circle on 3 shows that 3 is not a solution to x < 3.

Graph of the Inequapty x > 3

Here are the graphs of x > 3 and x ≥ 3, respectively

Graph of the Inequapty x ≥ 3

Graph of the Inequapty x ≠ 3

An inequapty with a “≠" sign has a solution set which is all the real numbers except a single point. Thus, to graph an inequapty with a " ≠ " sign, graph the entire pne with one point removed. For example, the graph of x ≠ 3 looks pke −

Graph the following inequapty on the number pne −

$frac{x}{3}$ ≤ 7

Solution

Step 1:

$frac{x}{3}$ ≤ 7; x ≤ (7 × 3); x ≤ 21

We first locate the point 21 on the number pne.

Step 2:

We put a closed circle on 21 and draw a thick pne towards left to denote the inequapty x ≤ 21

Graph the following inequapty on the number pne −

x > −32

Solution

Step 1:

We first locate the point −32 on the number pne.

Step 2:

We put a open circle on −32 and draw a thick pne towards right to denote the inequapty

x > −32