Linear Inequality Calculator

Linear Inequalities Calculator

Linear Inequality Formula

Formula

\[\text{If } X > Y:\quad X + Z > Y + Z,\quad X - Z > Y - Z \\ Z > 0:\quad X \cdot Z > Y \cdot Z,\quad \frac{X}{Z} > \frac{Y}{Z} \\ Z < 0:\quad X \cdot Z < Y \cdot Z,\quad \frac{X}{Z} < \frac{Y}{Z}\]

Where:

\(X\) and \(Y\) — the two values being compared (the left and right sides of the inequality)
\(Z\) — the number added to, subtracted from, multiplied by, or divided into both sides
The symbol \(>\) can be replaced by \(<\), \(\geq\), or \(\leq\); the same rules apply

These rules express the core property of inequalities: adding or subtracting any number preserves the direction of the inequality, while multiplying or dividing by a negative number reverses it.

Linear Inequality – Calculation Example

Example: start with the inequality \(5 > 3\) and add \(2\) to both sides.

\[ 5 > 3 \quad\Rightarrow\quad 5 + 2 > 3 + 2 \quad\Rightarrow\quad 7 > 5 \]

Result: \(7 > 5\). The direction of the inequality is unchanged, because addition never affects the sign.

Now multiply both sides of \(5 > 3\) by \(-2\) instead:

\[ 5 > 3 \quad\Rightarrow\quad 5 \cdot (-2) < 3 \cdot (-2) \quad\Rightarrow\quad -10 < -6 \]

Result: \(-10 \) to \(<\) because we multiplied by a negative number.

How to use the calculator

Select the Action you want to perform: Addition, Subtraction, Multiplication, or Division.
Enter the value X — the left side of the inequality (default: 5).
Enter the value Y — the right side of the inequality (default: 3).
Enter the value Z — the number applied to both sides (default: 2).
Read the Result, which shows three lines: the original inequality, the inequality after the transformation, and the final values.

What is a linear inequality?

A linear inequality is a mathematical statement that compares two expressions using one of the symbols \(>\), \(<\), \(\geq\), or \(\leq\), where the expressions involve variables raised only to the first power. Unlike an equation, which has a single exact solution, an inequality describes a range of values. For example, \(x > 4\) means every number greater than 4 — not just one number — satisfies the statement.

Solving inequalities works much like solving equations: you apply the same operation to both sides to keep the statement balanced. The crucial difference is the flip rule. When you multiply or divide both sides by a negative number, the direction of the inequality reverses. This happens because of how numbers are ordered on the number line: multiplying by a negative number reflects values across zero, so larger numbers become smaller ones. For instance, \(5 > 3\), but \(-10 < -6\).

This calculator demonstrates these transformations with concrete numbers. It takes a starting inequality \(X > Y\) or \(X < Y\) and shows exactly what happens to both sides when you add, subtract, multiply, or divide by \(Z\). Seeing the before-and-after comparison is one of the fastest ways to internalize the flip rule, which is the most common source of mistakes in algebra homework and exams.

Linear inequalities appear everywhere in real life: budget constraints (spending must be at most a certain amount), minimum requirements (a grade must be at least 60 to pass), and optimization problems in business and engineering. If you later study systems of inequalities or linear programming, the same one-sided rules extend naturally. To compare this with related topics, see the Bernoulli Inequality calculator or the Elimination Method calculator for solving systems of equations.

Assumptions, limitations, and practical tips

Negative multiplier or divisor: the calculator displays the transformed values with the original sign. When \(Z\) is negative in Multiplication or Division mode, always reverse the symbol yourself when interpreting the result — for example, if it shows \(-10 > -6\) after multiplying \(5 > 3\) by \(-2\), the correct statement is \(-10 < -6\).
Division by zero: entering \(Z = 0\) in Division mode returns an error, because dividing by zero is undefined.
Equal values: if \(X = Y\), there is no inequality to transform, and the calculator tells you so.
Strict vs. non-strict: the calculator works with strict inequalities (\(>\) and \(<\)). The same rules hold for \(\geq\) and \(\leq\), and the flip rule applies identically — the sign changes from \(\geq\) to \(\leq\) and back.
Fractions: division results are rounded to three decimal places; for exact fraction work, combine this tool with the Proportion calculator.
Remember the flip rule whenever you solve inequalities by hand: it applies only to multiplication and division by negatives — never to addition or subtraction, and never to positives.

Frequently Asked Questions

Why do you flip the inequality sign when multiplying by a negative number?

Multiplying by a negative number reflects values across zero on the number line, so the order reverses. For example, \(5 > 3\), but multiplying both sides by \(-2\) gives \(-10 < -6\). Adding or subtracting any number, or multiplying/dividing by a positive number, never changes the direction — only multiplication and division by negatives do.

How do I solve a linear inequality by hand step by step?

Treat it like an equation: isolate the variable by applying the same operation to both sides. For \(3x + 2 < 11\), subtract 2 to get \(3x < 9\), then divide by 3 to get \(x 6\) becomes \(x < -3\).

Does the flip rule apply to adding or subtracting a negative number?

No. The sign only reverses when you multiply or divide by a negative. Adding or subtracting any number — positive or negative — preserves the direction. So \(5 > 3\) stays \(5 + (-4) > 3 + (-4)\), which is \(1 > -1\). This is one of the most common mistakes in algebra homework.

What is the difference between strict and non-strict inequalities?

A strict inequality uses \(>\) or \( 4\) means 4 itself is not a solution, while \(x \geq 4\) means it is. The transformation rules are identical for both — even after flipping, \(\geq\) simply becomes \(\leq\).

Can I divide both sides of an inequality by zero?

No — division by zero is undefined for any number, so it is never a valid step, and this calculator returns an error for \(Z = 0\) in Division mode. If a variable could equal zero in a divisor, you must handle that case separately or use a different approach, such as moving all terms to one side and analyzing the sign.

What does the answer to an inequality mean, compared to an equation?

An equation has a single exact solution, but an inequality describes a whole range of values. When you solve \(x < 3\), the answer is every number less than 3, not just one number. On a number line you draw an open circle at 3 and shade everything to the left; a closed circle is used for \(\leq\) or \(\geq\).

Do the rules change if I take a reciprocal or square both sides?

Yes, be careful — those operations are not covered by the basic flip rule. Taking reciprocals of two positive numbers reverses the direction: \(4 > 2\) but \(\frac{1}{4} < \frac{1}{2}\). Squaring both sides is only valid when both sides are non-negative, and it may not preserve the inequality at all. Stick to addition, subtraction, multiplication, and division unless you verify these extra conditions.