Graphing Inequalities 

Graphing inequalities is what this lesson will teach you. It may help to remind you what an inequality is before showing how to graph it. An inequality is basically any mathematical statement that uses greater than (>), less than (<), less than or equal (≤), greater than or equal (≥), or not equal (≠) symbol. For example, the statements x > 6 and x ≤ -5 are examples of inequalities.

Graphing inequalities

Explore by building your own graph. Try the five classic examples, then test yourself.

Explore

Quick reference

  • > or < — open circle, the boundary number is not a solution.
  • ≥ or ≤ — closed (filled) circle, the boundary number is included.
  • ≠ — open circle, shade both directions (two rays, one point excluded).
  • Shade toward the values that make the statement true: right for "greater than", left for "less than".

Practice

x > 2
Score: 0 / 0

Graphing inequalities with five carefully chosen examples


Example #1

Graph x > 2

Draw a non-shaded or open circle at 2 and shade everything on the right of 2.

The shaded area in red is your solution. It means that the solution can be any number on the right of 2.

Notice that 2 is not shaded because 2 is not included in your solution. Notice also that although the last number on the number line is 9, your solution does not stop at 9.

Any other number bigger than 9 is a solution. In fact, the solution is infinite, so it would be very silly trying to graph all solutions to this problem.
Graph of x >2
Example #2

Graph x < -3

Draw an open circle at -3 and shade everything on the left of -3. Notice again that -3 is not shaded because -3 is not included in your solution. Notice again that although the last number on the number line is -9, your solution does not stop at -9.
Graph of x < -3
Example #3

Graph x ≥ 6

Draw a shaded circle at 6 and then shade everything on the right of 6. Notice that this time, the circle is shaded because x is also equal to 6.
Graph of x >= 6
Example #4

Graph x ≤ -1

Draw a shaded circle at -1 and then shade everything on the left of -1. Notice again that the circle is shaded because x is also equal to -1.
Graph of x <= -1
Example #5

Graph x ≠ 2

x ≠ 2 means that x is equal to any number except 2.

Draw a circle at 2 and shade everything on the right and on the left of 2.
Graph of x not equal to 2

You can also write x ≠ 2 as x > 2 or x < 2. Notice the use of the word "or" instead of "and". We could not use "and" because x cannot be bigger than 2 and be smaller than 2 at the same time.

Graphing equalities