Linear Inequality Solver
Solve single-variable linear inequalities (<, <=, >, >=) with step-by-step solutions and interval notation.
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In-depth guide for Linear Inequality Solver
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 $\ge$).
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.
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.
Mathematical accuracy verified by AptCalc Engine
Frequently Asked Questions (FAQs)
Answers to common questions about this calculator
What is a linear inequality?
How do you solve a linear inequality?
When do you flip the inequality sign?
How do you graph a linear inequality?
What is the difference between < and ≤?
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