Linear Inequalities Practice Test

Solve and interpret linear inequalities while controlling direction, endpoints, and special solution sets.

Linear Inequalities Practice Test

20 varied inequality questions with complete steps and endpoint explanations.

Instant feedback · Worked explanations

After the test · boundary & direction lab

A linear inequality describes a region of possible values, not one exact value

Solving an inequality uses many of the same operations as solving an equation, but the answer represents a set. That adds three important ideas: whether the boundary value is included, which direction the solution extends, and when the inequality sign must reverse.

< and >≤ and ≥Open endpointsClosed endpoints Reverse the signNumber linesCompound inequalitiesInterval notation

The most important exception to ordinary equation-solving habits: multiplying or dividing both sides by a negative number reverses the inequality sign.

less than

boundary excluded → open circle

greater than

boundary excluded → open circle

≤

less than or equal to

boundary included → closed circle

≥

greater than or equal to

boundary included → closed circle

Number-line gallery

The algebraic answer and the graph should tell the same story

x < 4

Open at 4, shade left.

x ≥ −2

−2

Closed at −2, shade right.

x ≤ 1

Closed at 1, shade left.

Direction-flip warning

A negative multiplier reverses order on the number line

Start −3x + 5 > 14

First remove the constant.

Subtract 5 −3x > 9

The sign has not changed yet.

Divide by −3 x < −3

Reverse > to <.

Check direction Use x = −4

−3(−4)+5 = 17, and 17 > 14.

Solve track · 4x − 7 ≤ 13

4x − 7 ≤ 13 The variable term has a positive coefficient.

4x ≤ 20 Add 7 to both sides.

x ≤ 5 Divide by positive 4, so the inequality direction stays the same.

✓

Boundary value x = 5 works 4(5) − 7 = 13, so the endpoint is correctly included.

Variables on both sides

Collect the variable terms just as you would in an equation — then watch the sign of the final coefficient

The direction changes only if the isolation step multiplies or divides by a negative quantity.

Original

5x + 2 > 8x − 10

Subtract 8x

−3x + 2 > −10

Subtract 2

−3x > −12

Divide by −3

x < 4

Compound inequality bands

“And” usually describes an overlap; “or” usually combines separate regions

AND · intersection

−2 < x ≤ 5

x must satisfy both conditions at the same time: greater than −2 and at most 5.

OR · union

x < −3 or x ≥ 4

Either region is allowed, so the graph has two separate shaded parts.

x < 4

(−∞, 4)

x ≥ −2

[−2, ∞)

−2 < x ≤ 5

(−2, 5]

x < −3 or x ≥ 4

(−∞, −3) ∪ [4, ∞)

Contextual limits

Inequalities naturally model minimums, maximums, budgets, capacities, and eligibility rules

Budget

12 + 4x ≤ 40

A fixed $12 fee plus $4 per unit cannot exceed $40.

Minimum score

x ≥ 70

70 is included because it is the minimum acceptable score.

Capacity

6x + 18 < 120

The total must remain strictly below 120.

Test-value scanner

A quick test value confirms both the boundary and the direction of the solution set

For x < 4, values below 4 should satisfy the inequality, while values at or above 4 should fail.

x = −2−2 < 4

belongs

x = 33 < 4

belongs

x = 44 < 4 is false

excluded

x = 77 < 4 is false

excluded

Boundary error scan

Most inequality mistakes involve direction, endpoint inclusion, or graph interpretation

Sign not reversed

−2x > 8 → x > −4 x < −4

Dividing by −2 reverses the inequality.

Wrong endpoint

x ≤ 5 drawn with an open circle Use a closed circle at 5.

The equality part of ≤ means the boundary belongs to the solution set.

Wrong shading

x < 4 shaded to the right Shade to the left.

Numbers less than 4 lie to the left of 4 on the number line.

AND vs OR

x < −3 or x ≥ 4 shown as one continuous interval Use two separate regions.

“Or” combines either allowed region rather than their overlap.

Inequality diagnostic spectrum

Sort missed questions by the visual or algebraic idea that failed

This separates ordinary algebra errors from inequality-specific direction and boundary errors.

Isolation Was the linear expression simplified and isolated correctly?

Direction Was the sign reversed after multiplying or dividing by a negative?

Boundary Was the endpoint included or excluded correctly?

Representation Did the number line, interval notation, and context match the algebraic solution?