Solving Inequalities

Solving an inequality means finding every value that makes a comparison true, not just one exact number. Reach for this when a problem sets a limit instead of an equality, such as $2x - 5 \le 9$, which asks for every $x$ keeping the left side at most $9$.

When You Use This Method

Use inequality solving whenever a problem describes a range, a cutoff, or a constraint rather than a single target value. An equation asks for the exact value(s) that make two sides equal; an inequality asks for all values that make one side larger, smaller, at least as large, or at most as large as the other.

For example, $x < 4$ means every number less than $4$ works, including $3$, $0$, and $-10$, but not $4$ itself. That is why inequality answers describe a set of numbers.

This shows up in score cutoffs, budget constraints, safety ranges, domain restrictions, and optimization, and throughout algebra in graphing intervals and compound inequalities.

The Steps

Solving an inequality follows the same isolating moves as an equation, with one extra rule.

1. Isolate the variable. Add, subtract, multiply, or divide both sides to get the variable by itself while keeping the statement equivalent.
2. Watch for negatives. If you multiply or divide both sides by a negative number, reverse the inequality sign.
3. Check the solution set. Read the final answer as a range of values, not a single number.
4. Test a value. Substitute a number from the solution set to confirm the original inequality is true.

The first move can be any equation-preserving operation: adding or subtracting the same number, or multiplying or dividing by the same positive number, all keep the inequality equivalent. Only multiplying or dividing by a negative number forces a sign flip:

The statement is still true, but only because the direction changed from $<$ to $>$.

Full Worked Example: Solve $-2x + 3 > 11$

Start the same way you would solve an equation by isolating $x$. Subtract $3$ from both sides:

Now divide both sides by $-2$. Because you are dividing by a negative number, reverse the inequality sign:

That is the full solution set: every number less than $-4$ makes the original inequality true.

The sign flips because negative numbers reverse order on the number line. If $a < b$, then $-a > -b$. You are not breaking the rules; you are preserving a true comparison after changing both sides by a negative factor.

Now run the test step both ways. A value inside the answer, $x = -5$:

Since $13 > 11$, it works. A value outside the answer, $x = 0$:

Since $3 > 11$ is false, that confirms the boundary at $x < -4$.

Where Each Step Trips People Up

At the isolating step, the most common error is forgetting to reverse the sign after multiplying or dividing by a negative number. If your answer points the "wrong way" against a quick test value, check this first.

At the solution-set step, students often treat the answer like a single value. For example, $x \ge 2$ means infinitely many values work, not just $x = 2$.

A subtler trap is dividing by a variable expression without knowing its sign. If the sign of that expression is unknown, the direction of the inequality may depend on a condition, so handle those cases separately. Use the final test step to catch any of these before trusting the answer.