How to Solve Inequalities

What changes when an equals sign becomes a less-than sign? Almost nothing in the steps, and one big thing in the answer. An equation usually gives one number, while an inequality gives a whole range of numbers. This guide shows how to solve inequalities of four types. It then turns budgets, grades, and weight limits into inequalities you can solve fast.

What Is the Step-by-Step Method to Solve Inequalities?

To solve an inequality, simplify both sides, move the x terms to one side, and isolate x with inverse operations. Flip the sign only when you multiply or divide by a negative number.

The moves match the ones in our guide to solving linear equations step by step. Here is the full method on -3x + 7 >= 19.

1. Subtract 7 from both sides: -3x >= 12.
2. Divide both sides by -3 and flip the sign: x <= -4.
3. Write the answer as a set: every number at or below -4.
4. Check with a point inside the answer and a point outside it.

Check With Test Points

Test x = -5, which sits inside the answer. The left side becomes -3(-5) + 7 = 22, and 22 >= 19 is true. Now test x = 0, outside the answer: 7 >= 19 is false.

Test the boundary too. At x = -4 the left side equals exactly 19, which confirms the endpoint belongs in the answer. Three quick checks catch almost every sign error before it costs you points.

Why Does the Inequality Sign Flip With a Negative?

Multiplying by a negative mirrors every number across zero on the number line. The mirror reverses left and right, so the number that was smaller becomes the larger one.

Picture 4 and 7 on a number line. The value 4 sits to the left, so 4 < 7. Multiply both by -1 and they land at -4 and -7. Now -7 is the one on the left, so -4 > -7.

The distance between the two numbers stays 3 units. Only their order reverses, and the inequality sign records order. That is why the sign must turn around to stay true.

Adding or subtracting never flips the sign. Adding 10 to both numbers slides them both right by 10, like a train moving along a track. The order stays the same, so 14 < 17 still holds.

The same reflection explains a less obvious case. Swapping the two sides of an inequality also reverses the symbol, so 5 < x means exactly the same thing as x > 5.

How Do AND and OR Compound Inequalities Differ?

An AND inequality keeps only the numbers that satisfy both parts, which is the overlap or intersection. An OR inequality keeps numbers that satisfy either part, which is the union.

Take the OR statement x – 4 < -6 or 2x + 1 > 9. Solve each piece on its own. The first gives x < -2, and the second gives x > 4. The answer is both rays together, with a gap from -2 to 4.

Now take the AND statement -1 <= 2x + 3 < 9. Subtract 3 from all three parts, then divide all three by 2. The result is -2 <= x < 3, a single closed-open segment.

Watch for a common trap. An AND statement whose parts never overlap, such as x < 1 and x > 5, has no solution at all. An OR statement like x < 5 or x > 1 covers every real number.

How Do You Solve Absolute Value Inequalities?

Read |x – a| as the distance from x to a. Then |x – a| < b means x lies within b units of a, and |x – a| > b means x lies farther away.

Solve |x – 5| < 3 with the distance idea. The numbers within 3 units of 5 run from 2 to 8, so 2 < x < 8. That is an AND statement written as -3 < x – 5 < 3.

Flip the direction to |x – 5| > 3. Now x must sit more than 3 units from 5, so x < 2 or x > 8. A greater-than absolute value always splits into an OR statement.

Test points confirm both answers. At x = 4 the distance is 1, and at x = 9 it is 4.

One shortcut saves time on tests. An inequality like |x – 5| < -2 has no solution, because a distance can never be negative. By the same logic, |x – 5| > -2 is true for every x.

How Do You Solve a Quadratic Inequality With a Sign Chart?

Find the roots of the quadratic, mark them on a number line, and test one point in each region. The regions with the correct sign form the answer.

Solve x^2 – 5x + 6 < 0. First factor it as (x – 2)(x – 3), which gives roots at 2 and 3. Finding roots is its own skill, covered in our guide on how to solve a quadratic equation.

The two roots split the line into three regions. Test x = 0 on the left: 0 – 0 + 6 = 6, which is positive. Test x = 2.5 in the middle: 6.25 – 12.5 + 6 = -0.25, which is negative. Test x = 4 on the right: 16 – 20 + 6 = 2, which is positive.

The inequality asks for negative values, so only the middle region works. The answer is 2 < x < 3. The roots stay out because the sign is strict, and the product equals 0 there.

How Do Real-World Limits Become Inequalities?

Translate limit words into symbols: “at most” becomes <=, “at least” becomes >=, and “no more than” becomes <=. Then solve and round in the direction the situation allows.

Rounding follows the real object. The elevator math gives b <= 20.44, but a partial box is impossible, so round down to 20. Twenty boxes total 980 kg, while 21 boxes reach 1,025 kg and break the limit.

Speed limits create AND statements. A highway with a 45 mph minimum and a 70 mph maximum means 45 <= s <= 70. To solve and graph your own linear or compound problem, try the free inequality solver with a number line.

FAQs About Solving Inequalities

Sources

This article covers linear, compound, absolute value, and basic quadratic inequalities in one real variable. Every worked example was checked by computation. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 26, 2026.