Graphing Inequalities on a Number Line

What an inequality says

An equation like x = 4 pins x to a single value. An inequality instead describes a range of values. The four symbols are:

Symbol	Meaning
<	less than
>	greater than
≤	less than or equal to
≥	greater than or equal to

So x > 3 means "x is any number bigger than 3", and x ≤ 5 means "x is 5 or any number below it". A number line is the clearest way to picture such a range. For a first look at the symbols, see inequalities.

Two things every graph shows

A number-line graph of an inequality has two parts:

1. A circle at the boundary number — open or filled.
2. An arrow (shaded line) showing which way the allowed values go.

Open or filled?

• Open circle (○) for < and >: the boundary number is not included.
• Filled circle (●) for ≤ and ≥: the boundary number is included.

A handy memory aid: the line under ≤ and ≥ means "and equal to", so you fill the circle in.

Which way does the arrow point?

• For x > a or x ≥ a, shade right (toward bigger numbers).
• For x < a or x ≤ a, shade left (toward smaller numbers).

Worked example 1: x > 2

Boundary is 2; symbol is >, so use an open circle. "Greater than" shades right.

        ○━━━━━━━▶
─┼──┼──┼──┼──┼──┼──┼─
 0  1  2  3  4  5  6

The open circle at 2 shows 2 itself is not a solution; everything to the right is.

Worked example 2: x ≤ −1

Boundary is −1; symbol is ≤, so use a filled circle. "Less than or equal to" shades left.

◀━━━━━━━●
─┼──┼──┼──┼──┼──┼─
−3 −2 −1  0  1  2

The filled circle at −1 shows −1 is included.

Worked example 3: solve first, then graph

Often you must solve the inequality before drawing it. You solve it exactly like an equation, using inverse operations on both sides.

Graph 3x + 1 ≥ 7.

3x + 1 ≥ 7
3x ≥ 6          (subtract 1 from both sides)
x ≥ 2           (divide both sides by 3)

Filled circle at 2, arrow to the right:

        ●━━━━━━━▶
─┼──┼──┼──┼──┼──┼─
 0  1  2  3  4  5

Worked example 4: the negative-flip rule

There is one special rule: if you multiply or divide both sides by a negative number, you must flip the inequality sign.

Solve and graph 8 − 2x > 2.

8 − 2x > 2
−2x > 2 − 8       (subtract 8 from both sides)
−2x > −6
x < 3             (divide by −2 and FLIP > to <)

Open circle at 3 (because of <), arrow to the left:

◀━━━━━━━○
─┼──┼──┼──┼──┼──┼─
 0  1  2  3  4  5

Why flip? Dividing by a negative reverses the order of numbers. Check with a value: x = 0 gives 8 − 0 = 8 > 2, true, and 0 is indeed less than 3. ✓

Worked example 5: a range (double inequality)

Graph −1 < x ≤ 3. This means x is greater than −1 and at most 3.

• At −1: open circle (<).
• At 3: filled circle (≤).
• Shade the segment between them.

   ○━━━━━━━━●
─┼──┼──┼──┼──┼──┼─
−2 −1  0  1  2  3

The integer solutions are the whole numbers covered: 0, 1, 2, 3. Note −1 is excluded because its circle is open.

A reliable method

1. If needed, solve the inequality using inverse operations (flip the sign if you ×/÷ by a negative).
2. Mark the boundary number with an open circle (<, >) or filled circle (≤, ≥).
3. Draw an arrow left for "less than", right for "greater than".
4. To list integer solutions, read off the whole numbers in the shaded region.

Activity: graph each inequality

Describe the circle (open/filled), its position, and the arrow direction. Then list any integer solutions where a range is given.

1. x < 4
2. x ≥ −3
3. 2x − 1 < 5
4. 5 − x ≤ 1
5. 0 ≤ x < 4 (list the integers)

Answers:

1. Open circle at 4, arrow left.
2. Filled circle at −3, arrow right.
3. Solve: 2x < 6 → x < 3. Open circle at 3, arrow left.
4. Solve: −x ≤ −4 → x ≥ 4 (flip when ÷ by −1). Filled circle at 4, arrow right.
5. Filled circle at 0, open circle at 4, shaded between. Integers: 0, 1, 2, 3.

Where this leads

Reading values off a line is the same thinking you use to plot points and read line graphs. Inequalities also appear later when shading regions on a coordinate grid, where each boundary is a straight line instead of a single point.