What “Is” Is Not

In mathematics, the word "is" does not always lead to an equal sign.

We use statements with boundaries all the time. A student must be at least sixteen to drive. A suitcase can weigh no more than fifty pounds. A temperature may stay below freezing. Each statement tells us what values are allowed.

That's the job of an inequality.

An inequality puts a boundary on what a value can be. Consider:

x > 5

The number 5 is the boundary. The statement tells us that x can be any number greater than 5. Instead of one answer, there are many values that make the statement true.

The wording also tells us whether the boundary belongs in the solution. If a person must be older than sixteen, sixteen does not qualify. If a person must be at least sixteen, it does.

That small difference changes the symbol:

x > 16 vs. x ≥ 16

Students often memorize that an open circle goes with one symbol and a closed circle goes with another. I would rather have them ask a question: Does the boundary work?

For x > 16, test 16. 16 > 16

That statement is false, so 16 is not included. The circle stays open.

For x ≥ 16, test 16 again. 16 ≥ 16

This statement is true, so 16 belongs in the solution. The circle is closed.

This way of thinking matters because inequality phrases can be easy to mix up. At least means the value can equal the boundary or go above it. At most means it can equal the boundary or stay below it. The words tell us where the limit is and which direction the values can go.

Students also need to understand that an inequality describes a set of possible values. Solving it does not usually produce one number. The answer may include every value to the left or right of a boundary on the number line.

When students see inequalities as boundaries, the symbols begin to make sense. The graph begins to make sense too. They are no longer trying to remember an isolated rule about circles and arrows. They are deciding which values make the statement true.

An inequality tells us how far “is” can go.