Linear Inequality Solver (ax + b ? c)

Understanding the linear inequality solver — concept and principle

A linear inequality such as ax + b < c asks not for one value but for the whole set of x-values that make the statement true. The answer is a half-line on the number line, such as x > −3. Inequalities describe limits and thresholds: the most items you can buy within a budget, the minimum score needed to pass, the temperature range a material can tolerate, or the break-even point above which a plan becomes profitable.

How it is calculated

For ax + b ? c, where ? is <, ≤, > or ≥:

1. Subtract b from both sides: ax ? c − b.
2. Divide both sides by a, giving x ? (c − b) ÷ a.
3. If a is negative, reverse the sign: < becomes >, ≤ becomes ≥, and vice versa.

Special case a = 0: the inequality becomes b ? c, which contains no x. If that statement is true (for example 4 < 10), every real x is a solution ("all real x"); if it is false, there is "no solution".

The principle

Adding or subtracting the same number on both sides keeps an inequality true, and so does multiplying or dividing by a positive number. Multiplying or dividing by a negative number reverses the order of the number line: 2 < 5, but −2 > −5. That is why the sign must flip when a < 0. Graphically, the solution set is the part of the line y = ax + b that lies below (or above) the horizontal line y = c; the boundary point x = (c − b)/a is where the two lines meet.

Worked example

1. Inputs: a = −2, b = 4, sign <, c = 10, so the inequality is −2x + 4 < 10.
2. Subtract 4 from both sides: −2x < 6.
3. Divide by −2. Because −2 is negative, < becomes >: x > 6 ÷ (−2).
4. Result: x > −3.
5. Check with a test value: x = 0 gives 4 < 10 (true); x = −5 gives 14 < 10 (false), so the solution set is correct.

How to read the result

x > −3 means every number greater than −3, but not −3 itself. On a number line it is drawn with an open circle at −3 and an arrow to the right; ≤ and ≥ use a filled circle because the boundary is included. In interval notation, x > −3 is (−3, ∞) and x ≥ −3 is [−3, ∞). In word problems, decide whether the boundary is allowed and whether only whole numbers make sense.

Common mistakes

• Forgetting to reverse the sign when dividing or multiplying by a negative number.
• Reversing the sign when subtracting a negative constant; only multiplication and division by a negative number flip it.
• Including the boundary point for a strict inequality (< or >).

Frequently asked questions

Why does the inequality sign flip for negative numbers?

Multiplying by a negative number reflects the number line around 0, so larger numbers become smaller. To keep the statement true, the direction of the sign must reverse.

How do I solve 3 < 2x + 1 ≤ 9?

Treat it as two inequalities, 2x + 1 > 3 and 2x + 1 ≤ 9, then combine: 1 < x ≤ 4.

What does "all real x" mean?

The inequality is true for every number, for example 0x + 4 < 10.

Sources: OpenStax, Elementary Algebra 2e, Section 2.7 Solve Linear Inequalities · I. M. Gelfand and A. Shen, Algebra, Birkhäuser