⚖️ Algebra Solver

Linear Inequality Solver

Q: How do you graph a linear inequality?

For a one-variable inequality, use an open circle for and a closed circle for ≤ or ≥. Shade the number line in the direction of the solution values.

Q: What is the difference between < and ≤?

< means less than and does not include the endpoint, while ≤ means less than or equal to and includes the endpoint.

Solve single-variable linear inequalities (<, <=, >, >=) with step-by-step solutions and interval notation.

Linear Inequality

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How to Use & Educational Guide

In-depth guide for Linear Inequality Solver

Educational Overview

What Is a Linear Inequality?

A linear inequality is an algebraic statement that compares two expressions using an inequality symbol such as $<$, $\le$, $>$, or $\ge$ instead of an equals sign. A linear inequality in one variable usually contains a variable such as $x$ raised to the first power and produces a range of possible solutions rather than one exact value.

For example, $2x + 4 < 10$ is a linear inequality. Solving it gives $x < 3$, which means every value less than $3$ satisfies the inequality.

Linear Inequality Formula

A linear inequality can be solved by isolating the variable in the same way as a linear equation, with one important exception: the inequality sign must be reversed when both sides are multiplied or divided by a negative number.

If $a > 0$: $ax + b < c \implies x < \frac{c-b}{a}$

If $a < 0$: $ax + b < c \implies x > \frac{c-b}{a}$

The same sign-reversal rule applies to all inequality symbols: $<$ becomes $>$, and $\le$ becomes $\ge$ when multiplying or dividing by a negative number.

How to Solve a Linear Inequality Step-by-Step?

To solve a linear inequality, use inverse operations to isolate the variable while keeping the inequality balanced.

Step 1: Simplify both sides. Remove parentheses and combine any like terms if necessary.
Step 2: Move constant terms. Add or subtract the constant from both sides to isolate the term containing the variable.
Step 3: Isolate the variable. Divide or multiply by the coefficient of the variable.
Step 4: Check the sign. If you divided or multiplied by a negative number, reverse the inequality symbol.
Step 5: Write the solution. Express the result as an inequality and, when useful, in interval notation.

Linear Inequality Symbols

Symbol	Meaning	Example
$<$	Less than	$x < 5$
$\le$	Less than or equal to	$x \le 5$
$>$	Greater than	$x > 5$
$\ge$	Greater than or equal to	$x \ge 5$

Linear Inequality Examples

Example 1: Solve $2x + 4 < 10$

• Step 1: Subtract $4$ from both sides: $2x < 6$.

• Step 2: Divide both sides by positive $2$: $x < 3$.

• Interval notation: $(-\infty, 3)$.

👉 Solution: $x < 3$

Example 2: Solve $-3x + 6 \ge 15$

• Step 1: Subtract $6$ from both sides: $-3x \ge 9$.

• Step 2: Divide by $-3$ and flip the inequality sign: $x \le -3$.

• Interval notation: $(-\infty, -3]$.

👉 Solution: $x \le -3$

Example 3: Variables on Both Sides: $5x - 8 > 2x + 7$

• Step 1: Subtract $2x$ from both sides: $3x - 8 > 7$.

• Step 2: Add $8$ to both sides: $3x > 15$.

• Step 3: Divide by $3$: $x > 5$.

• Interval notation: $(5, \infty)$.

👉 Solution: $x > 5$

Example 4: Inequality with a Fraction: $\frac{x}{4} + 3 \le 8$

• Step 1: Subtract $3$ from both sides: $\frac{x}{4} \le 5$.

• Step 2: Multiply both sides by positive $4$: $x \le 20$.

• Interval notation: $(-\infty, 20]$.

👉 Solution: $x \le 20$

How to Graph a Linear Inequality?

A one-variable linear inequality can be shown on a number line. Use an open circle when the endpoint is not included ($<$ or $>$), and a closed circle when the endpoint is included ($\le$ or $\ge$).

Open Circle: Used for $x < a$ or $x > a$ because $a$ is not included.

Closed Circle: Used for $x \le a$ or $x \ge a$ because $a$ is included.

For example, the solution $x > 5$ is graphed with an open circle at $5$ and shading toward the numbers greater than $5$.

Linear Inequalities in Interval Notation

Interval notation provides a compact way to represent the complete solution range.

$x < 3$ → $(-\infty, 3)$

$x \le 3$ → $(-\infty, 3]$

$x > 3$ → $(3, \infty)$

$x \ge 3$ → $[3, \infty)$

Real-World Applications of Linear Inequalities

💰 Budget Limits

If a product costs $15 and you have no more than $100 to spend, the inequality $15x \le 100$ can determine the maximum number of products you can purchase.

⚖️ Weight & Capacity Limits

Weight restrictions can be represented with inequalities such as $w \le 500$ kg, where the value cannot exceed the specified safety limit.

Common Mistakes When Solving Linear Inequalities

Forgetting to Flip the Sign: Always reverse the inequality when multiplying or dividing both sides by a negative number.
Using the Wrong Endpoint: Use an open endpoint for $<$ or $>$ and a closed endpoint for $\le$ or $\ge$.
Changing the Sign During Addition: Adding or subtracting a number does not require reversing the inequality sign.
Stopping Before Isolating the Variable: Continue simplifying until the variable is completely isolated.

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Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

What is a linear inequality?

A linear inequality is an algebraic comparison using symbols such as <, >, ≤, or ≥. A linear inequality in one variable produces a range of values that satisfy the condition.

How do you solve a linear inequality?

Simplify the inequality, isolate the variable using inverse operations, and reverse the inequality sign if you multiply or divide both sides by a negative number.

When do you flip the inequality sign?

You flip the inequality sign only when multiplying or dividing both sides by a negative number. For example, -2x > 6 becomes x < -3 after dividing by -2.

How do you graph a linear inequality?

For a one-variable inequality, use an open circle for < or > and a closed circle for ≤ or ≥. Shade the number line in the direction of the solution values.

What is the difference between < and ≤?

< means less than and does not include the endpoint, while ≤ means less than or equal to and includes the endpoint.