An equation asks: what value makes these two expressions equal? An inequality asks: for what values is one expression greater than (or less than) another? The answer is usually not a single value but an entire range of values.
>Greater than: x > 5 means x is strictly greater than 5. The value 5 itself is not included.
<Less than: x < 5 means x is strictly less than 5. Again, 5 is not included.
≥Greater than or equal to: x ≥ 5 means x can be 5 or any larger value. 5 is included.
≤Less than or equal to: x ≤ 5 means x can be 5 or any smaller value. 5 is included.
A helpful memory trick
The inequality sign always 'points to' the smaller value. In x > 5, the point aims at 5 , x is bigger. In x < 5, the point aims at x , x is smaller. The alligator always opens its mouth toward the larger number.
2 Solving Linear Inequalities
The process for solving inequalities is nearly identical to solving equations: use inverse operations to isolate the variable. The solution is a range of values, not a single answer.
Basic Inequality
Solve: x + 4 > 9
1Subtract 4 from both sides (same as equations): x + 4 − 4 > 9 − 4
2x > 5
3Solution: all real numbers greater than 5
Answer: x > 5
Two-Step Inequality
Solve: 3x − 2 ≤ 13
1Add 2 to both sides: 3x ≤ 15
2Divide both sides by 3: x ≤ 5
3Solution: all real numbers less than or equal to 5
Answer: x ≤ 5
3 The Critical Flip Rule
Here is the one rule that makes inequalities different from equations: when you multiply or divide both sides by a negative number, you must flip the inequality symbol.
The Flip Rule in Action
Solve: −2x < 8
1Divide both sides by −2. Because we're dividing by a negative, flip the sign.
2−2x ÷ (−2) > 8 ÷ (−2) ← symbol flipped from < to >
3x > −4
4Check: try x = 0 (which is > −4): −2(0) = 0 < 8 ✓. Try x = −5 (which is not > −4): −2(−5) = 10, is 10 < 8? No. The flip was correct.
Answer: x > −4
Never forget to flip!
−3x > 15 ÷ (−3) does not give x > −5. Dividing by −3 flips the sign: x < −5. This is the single most common error in solving inequalities. Always ask yourself: am I multiplying or dividing by a negative? If yes, flip.
Why does the sign flip?
Consider a true statement: 2 < 6. Multiply both sides by −1: −2 and −6. Is −2 < −6? No , −2 is greater than −6 on the number line. The direction reverses. The flip rule preserves the truth of the inequality after the operation.
4 Graphing on a Number Line
Inequality solutions are graphed as rays (half-lines) on a number line. Two conventions:
Graphing conventions
○Open circle (hollow dot) at the boundary value means that value is NOT included. Used for strict inequalities: > and <.
●Closed circle (filled dot) at the boundary value means that value IS included. Used for: ≥ and ≤.
For x > 3: draw an open circle at 3, then shade everything to the right (toward larger numbers). For x ≤ −1: draw a filled circle at −1, then shade everything to the left.
In interval notation: x > 3 is written (3, ∞). The parenthesis means "not included." x ≤ −1 is written (−∞, −1]. The bracket means "included."
5 Compound Inequalities
A compound inequality combines two inequalities. There are two types: AND (intersection) and OR (union).
AND Compound Inequality
Solve: −1 ≤ 2x + 3 < 9
1Treat this as two simultaneous conditions: 2x + 3 ≥ −1 AND 2x + 3 < 9
2Work on all parts simultaneously. Subtract 3 from all three parts: −4 ≤ 2x < 6
3Divide all parts by 2: −2 ≤ x < 3
4Solution: x is between −2 (inclusive) and 3 (exclusive)
Answer: −2 ≤ x < 3 (on a number line: closed circle at −2, open circle at 3, shaded between them)
OR Compound Inequality
Solve: x < −2 OR x ≥ 4
1These are two separate conditions , x satisfies the inequality if it meets EITHER one.
2x < −2: all numbers to the left of −2 (open circle at −2)
3x ≥ 4: all numbers 4 and to the right (closed circle at 4)
4The solution is everything to the left of −2 combined with everything 4 and above.
Answer: x < −2 or x ≥ 4 , two separate shaded regions on the number line
Try the Slope Calculator
Inequalities describe regions on a graph. The boundary of a linear inequality y > mx + b is the line y = mx + b , find its slope and intercept with our calculator.
Add 6: x > 8. Graph: open circle at 8, shade right.
Solve: −4x ≤ 20
Divide by −4, flip sign: x ≥ −5. Closed circle at −5, shade right.
Solve: 2x + 5 < 13
Subtract 5: 2x < 8. Divide by 2: x < 4. Open circle at 4, shade left.
Solve: −3 < x + 1 ≤ 5
Subtract 1 from all parts: −4 < x ≤ 4. Open circle at −4, closed circle at 4, shaded between.
Is x = −3 a solution to 2x + 1 > −4?
Substitute: 2(−3) + 1 = −5. Is −5 > −4? No. x = −3 is NOT a solution.
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.