Solve a linear inequality of the form ax + b (comparison) c and get the solution set for x.
📖 Linear Inequality AlgebraSolve for x: ax + b (?) c
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The inequality calculator solves a linear inequality in the form ax + b compared to c, and there's an important point to watch that differs from solving regular equations: when dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. For example, if we reach the step −2x < 6, dividing both sides by −2 means the result becomes x > −3. The calculator handles this case automatically based on the sign of coefficient a, and explicitly alerts the user when a reversal occurs. It also handles the special case where a=0, in which the inequality's validity depends only on comparing the two constant values b and c, regardless of x. This single sign-flip rule is the most common source of errors when solving inequalities by hand, which is exactly why this calculator makes the reversal step explicit rather than applying it silently. Checking the final answer by substituting a number from the solution range back into the original inequality is a reliable way to confirm the sign was handled correctly at every step.
Solving a linear inequality follows almost exactly the same steps as solving a linear equation, with one crucial exception that trips up even experienced students: dividing or multiplying both sides of an inequality by a negative number flips the direction of the inequality sign.
A concrete number check explains why this rule has to exist. Start with the true statement 2 < 5. Multiplying both sides by −1 gives −2 and −5. Without reversing the inequality, this would incorrectly claim −2 < −5, which is false — −2 is actually the larger number. Reversing the sign to −2 > −5 correctly preserves the true relationship. This is not an arbitrary convention layered on top of the math — it is a necessary correction to keep the inequality mathematically accurate whenever a negative multiplier is involved.
This rule has no equivalent when solving a regular equation, because equality is unaffected by sign changes — multiplying both sides of a true equation by any nonzero number, positive or negative, always preserves the equation's truth. Inequalities lack this symmetry, which is exactly why the reversal rule is unique to them and worth deliberately double-checking any time a negative coefficient appears in the solving process.
The special case where the coefficient of x becomes zero deserves separate attention, similar to the equation solver's special cases. If solving an inequality reduces to comparing two plain numbers with no x term remaining — for instance, arriving at 5 < 8 — the inequality is true for every possible value of x, since x has effectively dropped out of the problem entirely. If it instead reduces to a false numeric comparison like 8 < 5, the inequality has no solution at all, regardless of what value x might take.
Linear inequalities show up constantly in real decision-making: determining how many units of a product must be sold before a business becomes profitable (a break-even inequality), figuring out the maximum weight a vehicle or elevator can carry given a safety margin, or working out the range of acceptable input values for a formula to produce a result within a required range. In each of these cases, the answer is naturally a range of values rather than a single number, which is exactly what an inequality — rather than an equation — is built to express.
Writing the final answer clearly also matters: a solution like x > 4 is usually paired with a number-line sketch showing an open circle at 4 (since 4 itself does not satisfy the strict inequality) and shading extending in the correct direction, while a solution like x ≥ 4 uses a filled circle to show that 4 is included. Getting this notation right becomes especially important once inequalities are combined — describing a range like 2 < x ≤ 7 requires correctly tracking which endpoint is included and which is not, a detail that is easy to lose track of when solving several steps by hand.
Whenever you divide or multiply both sides of an inequality by a negative number, the direction of the inequality must reverse to keep the statement mathematically true.
With no x term left, the truth of the inequality depends only on comparing the constants b and c — the result is either true for every x or true for none.
Yes, if you choose ≤ or ≥ the boundary value is included in the solution; with strict < or > it is excluded.