Unit 8 · Lesson 8.1

One-variable inequalities

Real limits are rarely "exactly equal." You need at least $20 for the ticket, no more than 8 hours in the workday, a speed under the limit. Each of those describes a whole range of acceptable values, not one magic number, and that range is what an inequality is for.

Start with something you can picture: a sign at a ride that says "you must be at least 13." Is a 13-year-old allowed on? Yes. "At least 13" invites 13 itself in. Is a 20-year-old allowed? Yes. Is a 12-year-old? No.

So the rule doesn't pick out one age. It lets in a whole crowd of them, everyone from 13 upward. Write the age as a and that's a ≥ 13.

This is the first real difference from an equation. An equation like a = 13 has one answer. An inequality has many, and that's normal, not a sign something went wrong.

Each of the four signs is just a way of saying which numbers are let in:

• a < 13 means less than 13: everyone under 13, but not 13 itself.
• a > 13 means greater than 13: everyone over 13, but not 13 itself.
• a ≤ 13 means less than or equal to 13: under 13, and 13 too.
• a ≥ 13 means greater than or equal to 13: over 13, and 13 too.

The little line under ≤ and ≥ is doing real work: it's the "or equal to," and it's what invites the boundary number itself into the crowd. A sign without the line (< or >) leaves the boundary out. That one distinction shapes the picture you're about to draw, so read the sign slowly and ask, every time: is the boundary invited in, or not?

Here's the picture. The answer to an inequality lives on a number line, as a shaded stretch of it. You mark the boundary with a circle, and the circle tells you whether the boundary counts. An open circle (a hollow ○) sits at the boundary when the sign is strict, < or >, meaning the boundary itself is not included. A filled circle (a solid ●) sits there when the sign is inclusive, ≤ or ≥, meaning it is included. Then you shade the ray, the whole run of numbers that work, in the direction of the values that make the statement true. For x ≤ 2, that's a filled circle at 2 and shading running left, off toward the smaller numbers, because every number 2 and below belongs 8.1.f1.

Now the symbols. You solve an inequality almost exactly like an equation. Use the same balanced moves from Unit 2, whatever you do to one side you do to the other, to get the variable alone. Adding or subtracting from both sides never changes the direction. Multiplying or dividing both sides by a positive number never changes it either. There is one move, and only one, that does.

That one move: if you multiply or divide both sides by a negative number, the inequality sign flips. Less-than becomes greater-than, and the other way around. This is the rule that trips people, so don't take it on faith. Look at why it has to be true. Start with something nobody would argue with: $$2 < 3$$ Plainly true. Now multiply both sides by −1, which turns the 2 into −2 and the 3 into −3. Is −2 < −3? No. Owing $2 leaves you better off than owing $3, so −2 is the bigger number here. The statement 2 < 3 was true, and to keep it true after multiplying by the negative, the sign has to turn around: $$2 < 3 \;\xrightarrow{\;\times(-1)\;}\; -2 > -3$$ So the flip isn't a trick to memorize. It's the only way the statement stays honest when a negative reverses which side is bigger.

And here's the habit that makes this whole lesson safe: after you solve, test a point. Take any number from your answer set, put it back into the original inequality, and check that it makes a true statement. If it does, you can trust your answer.

And if you ever forget the flip, the test catches it. A number that clearly should work won't, and that mismatch is your signal to look again.

Read each worked example slowly, a line at a time, and ask why each line follows from the one before it. That's what turns a worked example into something you can actually reuse.