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Graphing Inequalities on a Number Line

Draw and read number-line graphs of inequalities: open or closed circle, left or right.

Practice this skill 20 min Graph solutions on a number line Leer en español
Key idea

What you'll learn

An inequality like x > 2 has infinitely many solutions: 3, 2.5, 2.001, 100, a million… You cannot list them all, so we draw them on a number line. In this lesson you will learn to:

  • graph the solution set of an inequality on a number line,
  • read a number-line graph and write the inequality it shows,
  • decide whether the boundary point gets an open or a closed circle,
  • graph "between" (compound) inequalities and whole-number solutions.

Why it matters on the GED: the test regularly shows four number lines and asks "Which graph shows the solution to 2x - 3 < 5?" Sometimes you must click the correct graph or the correct point (a hot spot item). Other times a graph is given and you choose the matching inequality. These questions are quick points once you know the two decisions every graph makes: which circle and which direction.

In everyday life, number-line thinking is how you picture limits: a speed limit of 65 mph (s \le 65), a minimum age of 18 (a \ge 18), or a fever above 100.4 °F (t > 100.4).

Words to know

Key words

  • Number line — a straight line with numbers spaced evenly; numbers get bigger to the right (example: 3 is right of -1).
  • Solution set — all values that make an inequality true (example: for x > 2, every number greater than 2).
  • Boundary point — the number where the solution set starts or ends (in x \le 4, the boundary is 4).
  • Open circle ○ — the boundary is not included; used for < and > (example: x > 2).
  • Closed (filled) circle ● — the boundary is included; used for \le and \ge (example: x \ge 2).
  • Ray / shading — the thick line with an arrow that shows all the solutions continue forever in one direction.
  • Compound inequality — two conditions at once, such as -2 \le x < 3 ("x is between -2 and 3").
  • Integer — a whole number or its negative: \ldots, -2, -1, 0, 1, 2, \ldots
  • Test point — a number you substitute to check which side of the boundary is shaded (example: 0 is an easy test point).

All Math words in the glossary

Two decisions: which circle, which direction

Every one-variable inequality graph comes from two decisions.

Decision 1 — the circle. Look at the symbol:

SymbolWordsCircle
<less thanopen ○
>greater thanopen ○
\leless than or equal toclosed ●
\gegreater than or equal toclosed ●

Memory trick: the line under \le and \ge "fills in" the circle. No line, no fill.

Decision 2 — the direction. When the variable is on the left:

  • x > \ldots or x \ge \ldots → shade to the right (bigger numbers).
  • x < \ldots or x \le \ldots → shade to the left (smaller numbers).

The symbol < even looks like an arrow pointing left when x is written first. That shortcut only works when x is on the left side, so rewrite first if needed.

The four basic graphs, all with boundary 2:

Four number lines from -2 to 6. x > 2: open circle at 2, shaded right. x >= 2: closed circle at 2, shaded right. x < 2: open circle at 2, shaded left. x <= 2: closed circle at 2, shaded left. x > 2 −2 −1 0 1 2 3 4 5 6 x ≥ 2 −2 −1 0 1 2 3 4 5 6 x < 2 −2 −1 0 1 2 3 4 5 6 x ≤ 2 −2 −1 0 1 2 3 4 5 6

Example A. Graph x \le -1. Closed circle at -1 (the symbol has a line under it), shade left.

Number line from -6 to 4. Closed circle at -1 with shading and an arrow to the left: the graph of x <= -1. x ≤ −1: closed circle, shade left −6 −5 −4 −3 −2 −1 0 1 2 3 4

Example B. Graph x > 3.5. Open circle halfway between 3 and 4, shade right.

Number line from -1 to 8. Open circle at 3.5 with shading and an arrow to the right: the graph of x > 3.5. x > 3.5: open circle, shade right −1 0 1 2 3 4 5 6 7 8 3.5

When the variable is on the right, or you must solve first

Variable on the right. 5 > x reads "5 is greater than x", which means "x is less than 5": x < 5. Turn the whole statement around — swap the sides and turn the symbol so it still opens toward the bigger side.

  • 5 > x is the same as x < 5 → open circle at 5, shade left.
  • -2 \le x is the same as x \ge -2 → closed circle at -2, shade right.
Two number lines. Top: 5 > x rewritten as x < 5, open circle at 5 shaded left. Bottom: -2 <= x rewritten as x >= -2, closed circle at -2 shaded right. 5 > x means x < 5 0 1 2 3 4 5 6 7 8 9 10 −2 ≤ x means x ≥ −2 −7 −6 −5 −4 −3 −2 −1 0 1 2 3

A common trap: someone sees ">" in 5 > x and shades right. Always rewrite with x first.

Solve before you graph. The GED usually gives an inequality that is not yet solved, like 6 - x > 2.

  1. Solve: subtract 6 → -x > -4; divide by -1 and flip → x < 4.
  2. Circle: < → open circle at 4.
  3. Direction: x < 4 → shade left.
Number line from -2 to 8. Open circle at 4 with shading and an arrow to the left: the graph of x < 4. Test value 0 is marked true (in the solution set). Test value 5 is marked false (not a solution). 6 − x > 2 → x < 4 −2 −1 0 1 2 3 4 5 6 7 8 x = 0 ✓ x = 5 ✗

If you forget to flip, you will shade the wrong way — and the GED nearly always includes that wrong graph as a choice.

Test-point check. After graphing, pick a number in the shaded part (0 is easiest if it is shaded) and put it in the original inequality. For 6 - x > 2, try x = 0: 6 > 2 ✓, and 0 is in the shaded part ✓. Then try a number on the unshaded side, such as x = 5: 1 > 2 is false ✓.

Between-graphs and whole-number solutions

Compound ("between") inequalities. The statement -2 \le x < 3 means both x \ge -2 and x < 3. Its graph is a segment, not a ray:

  • closed circle at -2 (because of \le),
  • open circle at 3 (because of <),
  • shading between them, no arrows.
Number line from -4 to 5. Closed circle at -2, open circle at 3, shaded between them: -2 <= x < 3. −4 −3 −2 −1 0 1 2 3 4 5

To read a segment graph, write: left number (symbol) x (symbol) right number, using \le for a closed circle and < for an open circle.

Counting integers in a segment. In -3 < x \le 2, the open circle at -3 is not included and the closed circle at 2 is. The integers are -2, -1, 0, 1, 2 — that's 5 integers.

Number line from -5 to 4. Open circle at -3, closed circle at 2, shaded between them: the graph of -3 < x <= 2. The integers -2, -1, 0, 1 and 2 are marked with dots: 5 integers. −3 < x ≤ 2: integers −2, −1, 0, 1, 2 (5 integers) −5 −4 −3 −2 −1 0 1 2 3 4 −3 out 2 in

Whole-number situations. When the variable counts things (people, tickets, boxes), only whole numbers make sense. Then the graph is a set of separate dots, not a solid ray. For "Kim can buy at most 5 shirts" (s \le 5, s \ge 0):

Number line from -1 to 7 with separate closed dots at 0, 1, 2, 3, 4 and 5 only: the whole-number solutions of x <= 5. −1 0 1 2 3 4 5 6 7

Work-backwards graphs. Sometimes a graph is given and you must find a missing number in the inequality. Read the boundary from the graph, set it equal to your solved boundary, and solve for the unknown. (See Worked example 5.)

Formula

Graphing rules to memorize (not on the formula sheet)

There is no formula, but memorize this summary — it is not on the GED formula sheet.

x > a \;\Rightarrow\; \circ \text{ at } a,\ \text{shade right} \qquad x \ge a \;\Rightarrow\; \bullet \text{ at } a,\ \text{shade right} x < a \;\Rightarrow\; \circ \text{ at } a,\ \text{shade left} \qquad x \le a \;\Rightarrow\; \bullet \text{ at } a,\ \text{shade left} a \le x < b \;\Rightarrow\; \bullet \text{ at } a,\ \circ \text{ at } b,\ \text{shade between}
  • a, b are the boundary numbers.
  • \circ = open circle (boundary not included), \bullet = closed circle (boundary included).
  • These rules assume x is written first (on the left). If it is not, rewrite it first.
Worked example

Worked example 1 (easy): graph a solved inequality

Problem: Describe the graph of x \ge -3.

  1. Circle: the symbol \ge has a line under it, so the circle is closed at -3.
  2. Direction: x is on the left and the symbol is "greater than", so shade right.

Answer: a closed circle at -3 with shading to the right.

Number line from -7 to 3. Closed circle at -3 with shading and an arrow to the right: the graph of x >= -3. Test value 0 is marked true (in the solution set). Test value -5 is marked false (not a solution). x ≥ −3 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 x = 0 ✓ x = −5 ✗

Check: 0 is in the shaded part, and 0 \ge -3 is true ✓. -5 is not shaded, and -5 \ge -3 is false ✓.

Worked example

Worked example 2: read a graph

Problem: A graph shows an open circle at 3 with shading to the right. Which inequality does it show?

Number line from -2 to 8. Open circle at 3 with shading and an arrow to the right: the graph of x > 3. Test value 4 is marked true (in the solution set). Open circle at 3, shaded right → x > 3 −2 −1 0 1 2 3 4 5 6 7 8 x = 4 ✓ 3 ✗
  1. Open circle → the boundary 3 is not included → the symbol is < or >.
  2. Shaded right → the solutions are bigger than 3 → "greater than".

Answer: x > 3

Check: 4 lies in the shaded region and 4 > 3 ✓; 3 itself sits in the open circle and 3 > 3 is false ✓.

Worked example

Worked example 3: solve, then graph (flip!)

Problem: Graph the solution of -2x \ge 6.

  1. Divide both sides by -2 and flip the sign: x \le -3.
  2. Circle: \le → closed circle at -3.
  3. Direction: x \le → shade left.

Answer: closed circle at -3, shaded left.

Two number lines from -8 to 2. Top, correct: -2x >= 6 solved with the flip gives x <= -3, closed circle at -3 shaded left. Bottom, wrong: forgetting the flip gives x >= -3, closed circle at -3 shaded right. Correct: −2x ≥ 6 → flip → x ≤ −3 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 Wrong (forgot to flip): x ≥ −3 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2

Check: Test x = -5 (shaded): -2(-5) = 10 \ge 6 ✓. Test x = 0 (not shaded): 0 \ge 6 is false ✓. A graph shaded to the right would come from forgetting the flip.

Worked example

Worked example 4 (real world): a weight limit

Problem: A small bridge has a sign: Weight limit 8 tons. Write an inequality for the allowed weight w (in tons) and describe its graph for real trucks.

  1. "Limit 8 tons" means the weight may be 8 tons or less: w \le 8.
  2. A weight cannot be negative or zero for a real truck, so w > 0.
  3. Together: 0 < w \le 8.
  4. Graph: open circle at 0, closed circle at 8, shading between them.

Answer: 0 < w \le 8 — open circle at 0, closed circle at 8, shaded between.

Number line from -1 to 10. Open circle at 0, closed circle at 8, shaded between them: the graph of 0 < w <= 8. Test value 5 is marked true (in the solution set). Test value 9 is marked false (not a solution). Bridge limit: 0 < w ≤ 8 tons −1 0 1 2 3 4 5 6 7 8 9 10 tons 5 t ✓ 9 t ✗

Check: A 5-ton truck (w = 5) is in the shaded part and may cross ✓; a 9-ton truck is outside and may not ✓.

Worked example

Worked example 5 (work backwards): find the missing number

Problem: The graph of the solution of k - 2x \le 1 is a closed circle at 3 shaded right. What is k?

Number line from -2 to 8. Closed circle at 3 with shading and an arrow to the right: the graph of x >= 3. Given graph: x ≥ 3 −2 −1 0 1 2 3 4 5 6 7 8
  1. The graph shows x \ge 3.
  2. Solve the inequality in terms of k. Subtract k: -2x \le 1 - k.
  3. Divide by -2 and flip: x \ge \dfrac{1 - k}{-2} = \dfrac{k - 1}{2}.
  4. The boundaries must match: \dfrac{k - 1}{2} = 3, so k - 1 = 6 and k = 7.

Answer: k = 7

Check: With k = 7: 7 - 2x \le 1 \Rightarrow -2x \le -6 \Rightarrow x \ge 3 ✓ — closed circle at 3, shaded right.

Calculator tip

Calculator tip (TI-30XS MultiView)

Graphing questions are about reading symbols and pictures, so the calculator is rarely needed. It helps only when you must solve a messy inequality first or test a value.

  • To test x = -5 in -2x \ge 6, type (−) 2 × (−) 5 enter → 10, then compare: 10 \ge 6 ✓. Use the (−) key for negative numbers.
  • For a fractional boundary like \tfrac{7}{2}, use the n/d key and press toggle to see 3.5, so you know where to place the circle (halfway between 3 and 4).

Don't use the calculator to decide the circle or the direction — those come from the symbol, and the calculator cannot help there.

Common mistake

Common mistakes

  • Open vs. closed mixed up. x \ge 3 drawn with an open circle. ✗ ✓ The line under \ge means the 3 is included → closed circle.
  • Wrong direction when x is on the right. 5 > x shaded right. ✗ ✓ Rewrite as x < 5 → shade left.
  • Forgetting to flip before graphing. -2x \ge 6 graphed as x \ge -3 (shaded right). ✗ ✓ Dividing by -2 flips: x \le -3 → shade left.
  • Including an open-circle endpoint when counting. In -3 < x \le 2, counting -3 as a solution gives 6 integers. ✗ ✓ Open circle means -3 is out: 5 integers.
  • Reading "left/right" as "negative/positive". A graph shaded right from -4 includes -3, -2, … — negative numbers can be in a "greater than" set. ✓ Right means bigger, not necessarily positive.
  • Using a solid ray for whole-number situations. If x counts people, the solutions are separate dots.
Test strategy

GED test strategy

  • Eliminate by circle first. Look at the symbol: if it is < or >, cross off every graph with a closed circle. Usually two choices disappear at once.
  • Then eliminate by direction. Pick a test point clearly on one side (0 is easiest) and check it in the original inequality. If it's true, the correct graph must shade that point.
  • Check the boundary number. Substitute it into the original inequality: the two sides should be equal. If not, the circle is in the wrong place.
  • For hot-spot items, click right on the circle or tick mark, not on the label below it.
  • Read the question twice: "graph the solution" vs. "which value is a solution" vs. "how many integers" are different tasks.
  • Time: a graph-matching item should take about a minute. Don't redraw all four graphs — eliminate.

Check yourself

Graph solutions on a number lineNo calculator

Which inequality is shown on the number line?

Number line from -1 to 7. Open circle at 3, shaded with an arrow to the right. −1 0 1 2 3 4 5 6 7
Answer choices
Hint

Look at two things: is the circle filled in, and which way does the shading go?

Check yourself

Graph solutions on a number lineNo calculator

Which description matches the graph of x \le -1 on a number line?

Answer choices
Hint

Does \le include the boundary? Which way are the numbers that are less than -1?

Summary

  • Open circle for < or > (boundary not included); closed circle for \le or \ge (boundary included).
  • With x written first: greater → shade right; less → shade left.
  • Rewrite 5 > x as x < 5 before graphing; solve (and flip if needed) before graphing.
  • "Between" inequalities like -2 \le x < 3 are segments with a circle at each end.
  • Check every graph with a test point in the original inequality.
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