Linear Inequality Solver
Solve linear inequalities of the form ax + b < c, ax + b > c, etc.
Quick Examples
What Is an Inequality?
An inequality is a mathematical statement that compares two expressions using a relational symbol instead of an equals sign. Rather than saying two values are the same, an inequality says one is greater than, less than, greater than or equal to, or less than or equal to another. The result is not a single point but a range of values that satisfy the condition.
What Is a Linear Expression?
A linear expression is an algebraic expression where every variable has an exponent of exactly 1 and no variables are multiplied together — for example, 3x + 7 or −2y + 5. There are no x², xy, or other non-linear terms. When a linear expression is compared to another value using an inequality symbol, the result is a linear inequality.
What Is a Linear Inequality?
A linear inequality uses a linear expression on one or both sides and replaces the equals sign with <, ≤, >, or ≥. The standard forms are:
The solution is always an interval or a set of intervals on the number line — every value of x that makes the statement true.
Everyday Inequality Examples
Inequalities appear in real life constantly, even when we don't write them symbolically:
- A speed limit sign says you must drive at most 60 mph — that's v ≤ 60
- A minimum age requirement of 18 years — age ≥ 18
- A bag can hold less than 10 kg — weight < 10
- A store requires a purchase of more than $50 for free shipping — total > 50
Equalities vs. Inequalities: Knowing the Difference
| Feature | Equality (=) | Inequality (<, ≤, >, ≥) |
|---|---|---|
| Solution type | One specific value | A range of values |
| Graph on number line | A single point | A ray or segment |
| Effect of ÷ by negative | No change needed | Inequality sign reverses |
| Notation | x = 4 | x > 4 or x ≤ 4 |
Types of Inequalities
1. Linear Inequalities in One Variable
The form is ax + b < c (or ≤, >, ≥). The solution is a single ray on the number line. Example: 2x − 3 > 7 → x > 5, written in interval notation as (5, ∞). This is the type handled by the calculator above.
2. Linear Inequalities in Two Variables
The form is ax + by < c. The solution is a half-plane on the coordinate grid — one side of the boundary line ax + by = c. If the symbol is strict (< or >), the boundary line is dashed; if inclusive (≤ or ≥), it is solid.
3. Quadratic and Polynomial Inequalities
These involve terms with exponents greater than 1, such as x² − 5x + 6 > 0. The solution is found by factoring, locating the roots, then testing sign intervals. The answer often consists of two separate intervals (e.g., x < 2 or x > 3).
4. Compound Inequalities
Two inequalities joined by and or or. An "and" compound (e.g., −3 < x ≤ 5) requires both conditions to hold — the solution is an intersection. An "or" compound (e.g., x < −1 or x ≥ 4) accepts either condition — the solution is a union.
5. Absolute Value Inequalities
These involve |ax + b| compared to a constant. Two rules apply:
- |expression| < k → −k < expression < k (a bounded interval)
- |expression| > k → expression < −k or expression > k (two separate rays)
Example: |x − 3| ≤ 4 → −4 ≤ x − 3 ≤ 4 → −1 ≤ x ≤ 7, written as [−1, 7].
How to Solve Inequalities Manually
The process mirrors solving equations, with one critical rule difference:
- Simplify both sides — distribute, combine like terms
- Move variable terms to one side by adding or subtracting
- Move constants to the other side by adding or subtracting
- Divide by the coefficient — if the coefficient is negative, reverse the inequality sign
- Write the solution in inequality notation or interval notation
Example: Solve −3x + 9 ≥ 0
Subtract 9: −3x ≥ −9 → Divide by −3 (reverse sign): x ≤ 3 → Solution: (−∞, 3]
Common Mistakes When Solving Inequalities
- Forgetting to flip the sign when multiplying or dividing by a negative number — the most frequent error
- Treating the solution like an equation — writing x = 3 instead of x ≤ 3
- Mixing up open and closed brackets in interval notation — use ( ) for strict < or >, and [ ] for ≤ or ≥
- Swapping the direction of a compound inequality — −2 < x < 5 means x is between −2 and 5, not outside that range
- Applying absolute value rules backwards — |x| < k is a bounded interval, not two rays
Linear Inequalities Examples
Example 1 — Basic one-variable: 4x + 8 < 20
Subtract 8: 4x < 12 → Divide by 4: x < 3 → Interval: (−∞, 3)
Example 2 — Negative coefficient: −5x ≥ 25
Divide by −5 (flip sign): x ≤ −5 → Interval: (−∞, −5]
Example 3 — Compound "and": −1 ≤ 2x + 3 ≤ 9
Subtract 3 throughout: −4 ≤ 2x ≤ 6 → Divide by 2: −2 ≤ x ≤ 3 → Interval: [−2, 3]
Example 4 — Absolute value: |2x − 1| > 5
Split: 2x − 1 > 5 or 2x − 1 < −5 → x > 3 or x < −2 → Interval: (−∞, −2) ∪ (3, ∞)
Interval and Set-Builder Notation Reference
Every inequality solution can be written three ways. This table covers the translations the calculator's answers rely on:
| Inequality | Interval Notation | Set-Builder Notation | Number Line |
|---|---|---|---|
| x < a | (−∞, a) | {x : x < a} | Open circle at a, ray to the left |
| x ≤ a | (−∞, a] | {x : x ≤ a} | Closed circle at a, ray to the left |
| x > a | (a, ∞) | {x : x > a} | Open circle at a, ray to the right |
| x ≥ a | [a, ∞) | {x : x ≥ a} | Closed circle at a, ray to the right |
| a < x < b | (a, b) | {x : a < x < b} | Open circles at a and b, segment between |
| a ≤ x ≤ b | [a, b] | {x : a ≤ x ≤ b} | Closed circles at a and b, segment between |
Two memory rules: round brackets ( ) mean the endpoint is excluded (strict < or >), and square brackets [ ] mean it is included (≤ or ≥). Infinity always takes a round bracket — it is a direction, not a number.
Quadratic Inequalities: The Case Table
For ax2 + bx + c compared to zero, the parabola's shape (set by the sign of a) decides everything. Find the roots r1 < r2 with the quadratic formula, then read the answer from this table:
| Parabola | ax2 + bx + c ≤ 0 | ax2 + bx + c ≥ 0 |
|---|---|---|
| a > 0 (opens upward) | Between the roots: [r1, r2] | Outside the roots: (−∞, r1] ∪ [r2, ∞) |
| a < 0 (opens downward) | Outside the roots: (−∞, r1] ∪ [r2, ∞) | Between the roots: [r1, r2] |
Worked example: Solve x2 − 5x + 6 > 0. Roots: x = 2 and x = 3. Since a = 1 > 0 the parabola opens upward, so the expression is positive outside the roots: x < 2 or x > 3, written (−∞, 2) ∪ (3, ∞). Check a test value, x = 0: 6 > 0 ✓
The Test-Point Method
Whenever you are unsure about a region, pick one easy number inside it and test the original inequality — one correct test settles the whole interval:
- Move everything to one side so the inequality reads "expression < 0" (or >, ≤, ≥).
- Find the roots — these are the boundary points that split the number line into regions.
- Choose one test point in each region and substitute it.
- Keep the regions that satisfy the inequality; discard the rest.
Worked example: Solve x2 − 9 ≤ 0. Roots: x = −3 and x = 3, giving regions x < −3, −3 < x < 3, x > 3. Test x = 0: 0 − 9 = −9 ≤ 0 ✓, so the middle region works. Test x = 4: 16 − 9 = 7 ≤ 0 ✗. Solution: [−3, 3].
Real-World Applications of Inequalities
- Budgeting: "The total must stay under $200" is a linear inequality in the quantities you buy.
- Manufacturing: Tolerances like "diameter must be 10 ± 0.05 mm" are compound inequalities.
- Sports: "Qualify with a time under 12 seconds" or "score at least 90 points".
- Profit zones: A quadratic profit function P(x) > 0 gives the production range that actually makes money.
- Safety limits: Maximum load, maximum dosage, minimum braking distance — all inequalities, not equations.