Graphing Inequalities on a Number Line Practice Test

Translate among inequalities, interval notation, endpoint symbols, shaded rays, compound graphs, and solution sets.

Graphing Inequalities on a Number Line Practice Test

20 number-line interpretation and graph-description questions with worked explanations.

Instant feedback · Worked explanations

Number line studio · graphing inequalities

After the test

Graphing an inequality on a number line requires three visual decisions: boundary, endpoint type, and direction

Once an inequality is already written with the variable isolated, the graph should be almost mechanical. Locate the boundary value, decide whether that value is included, and shade toward all numbers that satisfy the comparison. The graph is not decoration — it is another representation of the solution set.

Boundary valueOpen circleClosed circle Shade leftShade rightRead graph → inequality

Do not decide the shading direction from where the inequality symbol “points” on the page. Read the meaning: smaller values lie left, larger values lie right.

x < a or x > a

Open endpoint: a is excluded.

✓

x ≤ a or x ≥ a

Closed endpoint: a is included.

Direction ruler

Less-than solutions extend left; greater-than solutions extend right

x < 4

Open at 4; shade toward values smaller than 4.

x ≥ −2

−2

Closed at −2; shade toward values larger than −2.

x ≤ 5

Closed at 5; shade left because all values at or below 5 work.

Three-decision plotting workflow

Turn the inequality into a graph in the same order every time

Boundary Locate the number next to x.

For x ≥ −3, the boundary is −3.

Endpoint Decide open or closed.

≥ includes equality, so −3 gets a closed point.

Direction Shade the values that satisfy the inequality.

Greater than means values to the right.

Plotter examples

x > 2

x ≤ −1

−1

x < 6

Reversed-writing translator

If the variable is written on the right, rewrite the statement before graphing

4 > x

→

x < 4

Open at 4, shade left.

−2 ≤ x

→

x ≥ −2

Closed at −2, shade right.

7 ≥ x

→

x ≤ 7

Closed at 7, shade left.

Graph → inequality reader

You should be able to reconstruct the inequality from the endpoint and shading alone

x < 3

Open at 3 + shaded left.

−4

x ≥ −4

Closed at −4 + shaded right.

x ≤ 6

Closed at 6 + shaded left.

x < 5

Test x = 5 → 5 < 5 is false → open endpoint.

x ≤ 5

Test x = 5 → 5 ≤ 5 is true → closed endpoint.

x ≥ −2

Test x = −2 → −2 ≥ −2 is true → closed endpoint.

Number-line to interval bridge

Interval notation is another compact way to record the same one-direction solution set

x < 4

(−∞, 4)

4 is excluded, so use a parenthesis.

x ≤ 4

(−∞, 4]

4 is included, so use a square bracket.

x > −3

(−3, ∞)

Open endpoint at −3, extending right.

x ≥ −3

[−3, ∞)

Closed endpoint at −3, extending right.

Context to graph

Words such as “at least” and “no more than” determine the endpoint before you draw anything

Requirement

at least 16 years old

Variable

a = age

Inequality

a ≥ 16

Endpoint

closed at 16

Direction

shade right

Number-line error scanner

Most graphing mistakes are visual translation errors, not algebra errors

Wrong endpoint

x ≥ 4 drawn with an open circle. Use a closed circle at 4.

≥ includes the boundary.

Wrong direction

x < 4 shaded to the right. Shade left.

Numbers less than 4 lie to the left of 4.

Read symbol mechanically

4 > x interpreted as x > 4. 4 > x means x < 4.

Read the comparison in words before graphing.

Boundary misplaced

x ≤ −3 plotted at +3. Place the endpoint at −3.

Track the sign of the boundary value separately from the inequality symbol.

Number-line review console

Classify missed graphing questions by the exact visual decision that failed

This makes review more useful than simply repeating the same graphing exercise.

Boundary location Was the correct positive or negative value marked on the line?

Endpoint type Was equality translated into a closed endpoint when needed?

Shading direction Were smaller values shaded left and larger values shaded right?

Reverse reading Could you convert the finished graph back into the correct inequality?