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.