Compound Inequalities Practice Test

Solve intersections and unions, reverse signs correctly, use interval notation, and recognize empty or universal solution sets.

Compound Inequalities Practice Test

20 varied compound-inequality questions with instant explanations.

Instant feedback · Worked explanations

After the test · range builder

Compound inequalities combine two boundary conditions into one solution set

The central question is not just how to solve each inequality, but how the two solution sets relate. With AND, the answer is the overlap that satisfies both conditions. With OR, the answer includes values that satisfy either condition.

ANDORIntersectionUnion Double inequalityTwo branchesInterval notationNumber-line bands

Think in sets: AND narrows the allowed range; OR combines allowed regions.

AND · intersection x > −2 and x ≤ 5

Both statements must be true at the same time, so only values between −2 and 5 survive.

HOW
SETS
COMBINE

OR · union x < −3 or x ≥ 4

Either condition is enough, so the solution consists of two separate outer regions.

Range bands

The number line makes intersection and union visually obvious

−2 < x ≤ 5

−2 5

One continuous overlap between two boundaries.

x < −3 or x ≥ 4

gap

Two separated solution regions; the middle gap is excluded.

Solve a three-part inequality · −5 < 2x + 1 ≤ 9

START

−5 < 2x + 1 ≤ 9 The middle expression must satisfy both boundaries simultaneously.

−1

−6 < 2x ≤ 8 Subtract 1 from all three parts.

÷2

−3 < x ≤ 4 Divide all three parts by positive 2; directions stay unchanged.

SET

(−3, 4] Open at −3, closed at 4.

AND branch solver

When the compound statement is written as two separate inequalities, solve both and keep only the overlap

Branch A

2x − 3 > 1
2x > 4
x > 2

AND

Branch B

x + 4 ≤ 9
x ≤ 5

Combined solution: 2 < x ≤ 5, or in interval notation, (2, 5].

OR branch solver

For OR, solve both branches and keep both valid regions

Branch A

3x + 2 < −7
3x < −9
x < −3

Branch B

2x − 1 ≥ 7
2x ≥ 8
x ≥ 4

Combined solution: x < −3 or x ≥ 4, or (−∞, −3) ∪ [4, ∞).

Compound interval notation

Brackets show included boundaries; parentheses show excluded boundaries

−2 < x ≤ 5

(−2, 5]

One connected interval: open on the left, closed on the right.

1 ≤ x < 7

[1, 7)

Closed at 1 and open at 7.

x < −3 or x ≥ 4

(−∞, −3) ∪ [4, ∞)

Two separate intervals joined by the union symbol.

Negative-multiplier checkpoint

If all three parts of a double inequality are divided by a negative number, both inequality signs reverse

Original

−12 ≤ −3x < 6

Divide by −3

4 ≥ x > −2

Rewrite in order

−2 < x ≤ 4

Range constraint model

Compound inequalities naturally describe values that must stay between a minimum and maximum

Allowed temperature

18°C to 24°C

Lower bound

T ≥ 18

Upper bound

T ≤ 24

Compound form

18 ≤ T ≤ 24

Interval

[18, 24]

Compound inequality error scan

The main mistakes come from mixing up intersection, union, and three-part operations

AND treated as OR

x > 2 and x ≤ 5 → x > 2 or x ≤ 5 Keep only the overlap: 2 < x ≤ 5.

AND requires both conditions simultaneously.

OR forced into one interval

x < −3 or x ≥ 4 → (−3, 4) (−∞, −3) ∪ [4, ∞)

OR can produce disconnected solution regions.

Middle only changed

−5 < 2x + 1 ≤ 9 → −5 < 2x ≤ 9 after subtracting 1 −6 < 2x ≤ 8

Apply the operation to all three parts.

Negative division

−12 ≤ −3x < 6 → 4 ≤ x < −2 −2 < x ≤ 4

Both inequality signs reverse when dividing all parts by −3.

Range-builder diagnostics

Sort missed questions by the set operation or boundary rule that failed

This keeps review focused on compound inequalities rather than general inequality solving.

AND / intersection Could you identify the region satisfying both conditions?

OR / union Could you preserve both valid branches without forcing them into one interval?

Three-part operations Was each operation applied to the left, middle, and right parts?

Endpoints & notation Did the graph and interval notation match strict versus inclusive bounds?