How to Solve Inequalities Step by Step

An inequality compares values using signs such as less than, greater than, less than or equal to, and greater than or equal to. Suppose you need to solve x + 4 > 9. Subtract 4 from both sides to get x > 5. Every number greater than 5 satisfies the original statement.

You can use the same balancing idea as an equation. If 3x <= 18, divide both sides by 3 to get x <= 6. Adding, subtracting, multiplying, or dividing both sides by the same positive number preserves the comparison.

This is the main method for solving inequalities: simplify, perform inverse operations, and keep the two sides balanced. The answer is usually a range of possible values rather than one single number. You can often check the direction by asking whether a larger or smaller value should make the original statement true.

The special rule appears when you multiply or divide by a negative number. For example, solve -2x > 8. Dividing both sides by -2 gives x < -4. The greater-than sign reverses and becomes less than.

Why does the sign flip? Start with a true statement such as 3 > 1. Multiply both sides by -1 and you get -3 < -1. On a number line, multiplying by a negative reflects values across zero, which reverses their order.

This rule is the most common mistake in solving inequalities. The sign does not reverse because you moved a term across the equals area or because a negative number appeared somewhere in the question. It reverses only when both sides of the inequality are multiplied or divided by a negative quantity.

A number line makes an inequality answer visible. For x > 5, place an open circle at 5 because 5 itself is not included, then shade to the right. For x <= 6, use a filled circle at 6 because equality is allowed, then shade to the left.

You can check your algebra by choosing a value from the shaded region and substituting it into the original inequality. If x < -4, try x = -5 in -2x > 8. The left side becomes 10, and 10 > 8 is true. Then try a value outside the solution, such as 0. The statement becomes 0 > 8, which is false.

Compound inequalities can describe a range between two boundaries. For example, 2 < x <= 7 means x is greater than 2 and at most 7. You can show both boundaries on one number line and shade only the values that satisfy both conditions. If a graph is required, label the boundary clearly before shading. If interval notation is required, translate the same boundary rules carefully so the algebra and graph still describe the same set of values. When solving inequalities, write one algebra step per line. That makes it easier to spot the exact moment a negative division happens and prevents a silent sign error.

Solving inequalities is mostly about the same balanced operations used for equations. Isolate the variable, but reverse the inequality sign whenever you multiply or divide both sides by a negative number. Use a number line and a quick substitution check to confirm that your solution range points in the correct direction.