How to Solve Linear Inequalities

Quick answer

To solve linear inequalities, first classify the problem, preserve the meaning of every transformation, and verify the final result in the original statement. For the worked example -2x + 3 > 11, the result is x < -4. The method below shows why that result follows rather than presenting it as an unexplained calculator output.

Identify the problem before calculating

Linear inequalities use the same balancing operations as linear equations, except multiplying or dividing by a negative reverses the inequality direction. This classification step matters because a familiar-looking expression can require a different rule when its structure changes. Write down any domain restriction, unit, dimension, or reference value before manipulating the symbols.

Keep every transformation reversible. An algebraic line is not merely formatting: it is a claim that the solution set has not changed. Record restrictions before cancelling factors or applying inverse operations.

Step-by-step method

1. Simplify both sides and combine like terms.
2. Move variable terms and constants as needed.
3. When dividing by a negative number, reverse <, >, ≤, or ≥.
4. Write the solution in inequality and interval notation.
5. Test one value inside and one value outside the proposed set.

Each line should answer two questions: what operation was performed, and why that operation preserves or correctly changes the mathematical object. When a calculator or browser tool is used, keep the original input visible so the output can be compared with the source problem.

Worked example

1. Start with -2x + 3 > 11.
2. Subtract 3 to get -2x > 8.
3. Divide by -2 and reverse the sign: x < -4.
4. A test value x=-5 works, while x=0 does not.

The result x < -4 is not accepted solely because it looks plausible. The example retains the intermediate quantities that make a sign, denominator, exponent, dimension, or rounding mistake visible.

Read the visual explanation

The three diagrams on this page separate the concept, the execution sequence, and the final check. Use the concept diagram to recognize the structure, the workflow diagram while solving, and the verification diagram after obtaining a candidate result. The diagrams are SVG files, so text remains selectable and the graphics stay sharp on mobile screens and when printed.

Common mistakes

• Forgetting to reverse the sign after division by a negative.
• Using a closed endpoint for a strict inequality.
• Treating an inequality solution as one number instead of a set.

A reliable correction strategy is to return to the earliest line where the mathematical object changed. Repeating only the final arithmetic often reproduces the same hidden setup error.

Verify the result

Substitute x=-5: -2(-5)+3=13>11. Substitute x=0: 3 is not greater than 11. Also inspect whether the answer has the correct sign, scale, units, dimension, and domain. For an equation, substitution is decisive. For an inverse pair such as differentiation and integration, apply the inverse operation. For a statistical or financial result, recompute one intermediate quantity and compare it with an independent method.

Use the tool without losing the reasoning

The Equation Solver deliberately does not claim to solve inequalities. Verify the interval with one test value inside it and one outside it, and confirm that dividing by a negative number reversed the inequality sign. This keeps the guide accurate instead of sending an unsupported expression to a calculator.

Keep a compact record containing the original problem, the selected method, intermediate values, the final result, and the verification. That record is more useful for study, teaching, and debugging than an isolated answer.

Visual explanation