Solving Inequalities

Solving inequalities means finding all values that make a comparison true. If you can solve a basic equation, you already know most of the process. The main extra rule is this: if you multiply or divide both sides by a negative number, reverse the inequality sign.

For example, $2x - 5 \le 9$ asks for every value of $x$ that keeps the left side less than or equal to $9$. The answer is usually a range of values, not one exact number.

What Solving An Inequality Means

An equation asks for 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. That includes $3$, $0$, and $-10$, but not $4$ itself. That is why inequality answers describe a set of numbers.

Rules For Solving Inequalities

These steps keep an inequality equivalent:

• Add the same number to both sides.
• Subtract the same number from both sides.
• Multiply both sides by the same positive number.
• Divide both sides by the same positive number.

If you multiply or divide both sides by a negative number, reverse the sign:

$$3 < 5$$

Multiply both sides by $-1$:

$$-3 > -5$$

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

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

Start the same way you would solve an equation: isolate $x$.

Subtract $3$ from both sides:

$$-2x > 8$$

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

$$x < -4$$

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

Why The Inequality Sign Flips

Negative numbers reverse order on the number line. If $a < b$, then $-a > -b$.

That is why dividing $-2x > 8$ by $-2$ turns the answer into $x < -4$ instead of $x > -4$. You are not breaking the rules. You are preserving a true comparison after changing both sides by a negative factor.

Check The Answer With One Value

Test a value that fits the answer, such as $x = -5$:

$$-2(-5) + 3 = 13$$

Since $13 > 11$, the original inequality is true.

Now test a value outside the answer, such as $x = 0$:

$$-2(0) + 3 = 3$$

Since $3 > 11$ is false, that matches the solution $x < -4$.

Common Mistakes When Solving Inequalities

The most common mistake is forgetting to reverse the sign after multiplying or dividing by a negative number.

Another mistake is treating the answer like a single value instead of a range. For example, $x \ge 2$ means infinitely many values work, not just $x = 2$.

A third mistake 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.

Where Solving Inequalities Is Used

Inequalities appear whenever a problem has limits instead of exact equality. Common examples include score cutoffs, budget constraints, safety ranges, domain restrictions, and optimization problems.

They also show up throughout algebra, especially in graphing intervals, solving compound inequalities, and describing feasible solutions in real situations.

Try A Similar Inequality

Try solving $5x - 7 \le 18$. Then solve $-3x + 4 \ge 1$ and compare the last step. If you want to go further, try your own version with a negative coefficient and check whether you flipped the sign at the right moment.