Inequality Solver

About Inequality Solver

When dividing or multiplying an inequality by a negative number, the inequality sign flips direction.

How to use this calculator

1. Enter the coefficient \(a\), the constant \(b\), and the constant \(c\) from your inequality.
2. Make sure your inequality is in the form \(ax + b < c\).
3. Click solve to isolate \(x\) and get the solution.
4. Read the result as all values of \(x\) that make the inequality true.

The formula explained

The formula \(ax + b < c \Rightarrow x < \frac{c-b}{a}\) finds the values of \(x\) that satisfy the inequality after undoing the addition or subtraction of \(b\) and dividing by \(a\). It gives the boundary value, then tells you which side of that value works.

• a = the coefficient of \(x\)
• b = the constant added to \(ax\)
• c = the constant on the right side of the inequality
• x = the variable you are solving for

Step by step method

1. Start with the inequality \(ax + b < c\).
2. Subtract \(b\) from both sides to get \(ax < c-b\).
3. Divide both sides by \(a\) to isolate \(x\), giving \(x < \frac{c-b}{a}\) when \(a > 0\).

Worked example

Problem. Solve the inequality \(3x + 5 < 20\).

1. Subtract 5 from both sides: \(3x < 15\).
2. Divide both sides by 3: \(x < 5\).
3. Check: any number less than 5 makes \(3x + 5\) less than 20.

Tips and common mistakes

• If \(a\) is negative, the inequality sign must flip when you divide by \(a\). For example, dividing by \(-2\) turns \(<\) into \(>\).
• The answer is often an interval of numbers, so be careful not to write only one value when the solution includes many values.

Frequently asked questions