What is a linear inequality?
A linear inequality compares two first-degree expressions with <, >, ≤, or ≥. Instead of usually producing one value, it describes a set of values. For x > 3, every real number greater than 3 works. The boundary 3 is excluded because the symbol is strict.
The symbols ≤ and ≥ include equality, so their boundary points belong to the solution set. On a number line, strict endpoints use an open circle and inclusive endpoints use a closed circle. Interval notation records the same distinction with parentheses and brackets.
How to solve a linear inequality step by step
Use distribution, combining like terms, and inverse operations as you would for an equation. Keep the comparison symbol unchanged when adding or subtracting the same value from both sides. Watch the sign of the number used in the final multiplication or division.
- Simplify each side by distributing and combining like terms.
- Collect variable terms on one side and constants on the other.
- Add or subtract the same expression on both sides without changing the inequality direction.
- Divide by the coefficient of x; reverse the symbol only if that coefficient is negative.
- State the solution as an inequality and, when useful, in interval notation.
- Check the boundary and a representative value from the claimed solution set.
Why does the inequality sign reverse for a negative number?
Multiplication by a negative reflects numbers across zero and reverses their order. Since 2 < 5, multiplying both values by −1 produces −2 > −5. The same rule applies when dividing by a negative because division by −a is multiplication by −1/a.
The sign does not reverse merely because a negative number appears somewhere in the line. It reverses at the exact step where both sides are multiplied or divided by a negative quantity. A good inequality calculator calls out that moment explicitly.
Worked example: −3x + 6 > 15
Subtract 6 from both sides to get −3x > 9. Now divide both sides by −3. Because the divisor is negative, reverse > to <: x < −3. In interval notation, the answer is (−∞, −3), and −3 is excluded.
Check x = −4, a value inside the interval: −3(−4) + 6 = 18, and 18 > 15 is true. Check the boundary x = −3: 9 + 6 = 15, and 15 > 15 is false, so the open endpoint is correct.
How to read interval notation
A bracket includes a finite endpoint, while a parenthesis excludes it. Infinity is not a number that can be included, so ±∞ always uses a parenthesis. Thus x ≥ 2 becomes [2, ∞), and x < −1 becomes (−∞, −1).
The calculator currently focuses on supported one-variable linear inequalities. Compound inequalities, absolute-value inequalities, and rational inequalities may require additional interval analysis beyond the published capability.
Common inequality mistakes
The most frequent mistake is forgetting the reversal after division by a negative. Other errors include using a closed endpoint for < or >, reversing after addition of a negative, and checking only the boundary even though a strict boundary is expected to be false.
A single test point does not replace the algebraic derivation, but it can expose a reversed interval quickly. Choose a simple value clearly inside the proposed set and evaluate the original inequality.