Solving and Graphing One-Variable Inequalities

In this lesson you will: Solve a linear inequality using the same moves you already use on equations; Graph a solution set on a number line, with an open or a filled circle; Test a value to confirm that your solution set is the right one.

The answer is a set, not a number

Solve 3x+5=20 and you get one value, x=5. Solve 3x+5<20 and you get x<5 — every number below five, and there are infinitely many of them.

That is the only real difference. The steps that got you from 3x+5=20 to x=5 get you from 3x+5<20 to x<5, because subtracting 5 from both sides and dividing both sides by 3 leave the order of the two sides alone.

Four symbols, two kinds of circle

The circle at the boundary records whether the boundary number is itself a solution. Strict symbols leave it out; the or equal to symbols include it.

Read the symbol out loud in words before you draw. x >= -3 is \"x is at least negative three,\" and \"at least\" includes the number itself.

Every move you already know still works

Add the same number to both sides, subtract the same number, multiply both sides by a positive number, divide by a positive number — all four keep the inequality pointing the same way.

There is exactly one exception in the whole subject, and it gets its own lesson. Multiplying or dividing by a negative number reverses the sign. Everything else behaves.

Common mistake — Answering with the boundary number instead of the set

Asked to solve 3x+5<20, a student writes x=5.

It is a reasonable slip, because the arithmetic really does end at a 5, and years of solving equations have trained the hand to write an equals sign at the last line.

But 5 is the one number in the neighborhood that is not a solution: 3(5)+5=20, and 20<20 is false. The boundary is where the solutions stop, not where they are.

The same habit produces a filled circle on a graph that should have an open one.

How to catch it: Substitute the boundary number into the original inequality. If it makes a true statement, the circle is filled and the symbol is <= or >= ; if it makes a false one, the circle is open. Either way the final answer names a range, so if your last line contains an equals sign, look again.