Inequalities: learn and practice

Who this lesson is for

This inequalities lesson is for grade 8. Students solve a linear inequality such as 3x − 5 > 7 and report a range, not a single number. The catalog skill continues into grade 9. Published hubs stop at grade 8, so only the grade 8 hub links here. The practice link opens that skill.

Balance, then keep the direction

You may add or subtract the same amount on both sides, just as in an equation. You may divide by a positive number, and the inequality sign stays. The catalog exercises use a positive coefficient of x, so the sign you start with is the sign you finish with. The solution is a set of numbers. x > 4 means every number greater than 4, not the single number 4.

The boundary is not always included

A strict sign, > or <, does not include the boundary. In 3x − 5 > 7 the value x = 4 makes both sides equal, and equal is not greater. So 4 itself is not a solution. A sign of ≥ or ≤ would include the boundary. Read the sign in the problem before you copy it into the answer. Dropping it and writing only 4 turns the inequality back into an equation.

Worked example 1: 3x − 5 > 7

Solve 3x − 5 > 7. Add 5 to both sides: 3x > 12. Divide by 3. The coefficient is positive, so the sign stays: x > 4. Check a number inside the range and a number outside it. x = 5 gives 3 × 5 − 5 = 10, and 10 > 7, so 5 works. x = 4 gives 7, and 7 is not greater than 7, so 4 does not work. This is the catalog exercise. The answer is x > 4, not x = 4.

Worked example 2: 2x + 1 < 9

Solve 2x + 1 < 9. Subtract 1 from both sides: 2x < 8. Divide by 2. The coefficient is positive, so the sign stays: x < 4. Check x = 3: 2 × 3 + 1 = 7, and 7 < 9. Check x = 4: 2 × 4 + 1 = 9, and 9 is not less than 9. The boundary fails, which matches the strict sign. The solution is every number less than 4.

Worked example 3: 4x > 12

Solve 4x > 12. There is no added term to undo. Divide both sides by 4: x > 3. The sign stays because 4 is positive. Check x = 3: 4 × 3 = 12, which is not greater than 12. Check x = 4: 16 > 12, so 4 is in the solution. Writing x = 3 solves the equation 4x = 12. The inequality asks for the numbers that make the left side larger than the right side.

Checks before you leave the problem

Undo addition before you divide, and keep the sign when you divide by a positive number. Test one value that should work and the boundary that should not, when the sign is strict. Then practice inequalities on the board.

Common questions about inequalities

Why is the answer not a single number?

An inequality describes a range. 3x − 5 > 7 is true for every x greater than 4. The equation 3x − 5 = 7 would be the one with the single solution x = 4.

When does the inequality sign flip?

It flips when you multiply or divide both sides by a negative number. The exercises linked from this page use a positive coefficient, so the sign stays. The catalog hint says the same thing.

Is x = 4 a solution of x > 4?

No. The sign > does not include the boundary. x = 4 makes the two sides equal, and the statement asked for greater.