Inequalities: when one side is bigger

An equation asks a narrow question: which single value makes the two sides match? An inequality asks something more honest about the real world — not "what number is it?" but "how big does it need to be?" The bus holds at most 40 people. You need at least $12 for lunch. Your score has to beat 85. None of these pin down one answer; each one carves out a whole range of answers. Learning to read and solve inequalities is learning to think in ranges.

Four symbols, four shades of meaning

Everything in this lesson rests on four little symbols. The trick is to read the open end of the symbol as "the bigger side."

• \(a < b\) means \(a\) is strictly less than \(b\). The point of the symbol aims at the smaller number.
• \(a > b\) means \(a\) is strictly greater than \(b\).
• \(a \le b\) means \(a\) is less than or equal to \(b\) — the line underneath quietly adds "or exactly equal."
• \(a \ge b\) means \(a\) is greater than or equal to \(b\).

That little underline is not decoration. "You must be over 18" (strict) and "you must be at least 18" (allows exactly 18) are different rules, and they let in different people. We will keep returning to that distinction, because it decides one crucial detail when we draw the answer.

Solve them just like equations — almost

Here is the good news. You already know how to do most of this. The moves you learned for equations — add the same thing to both sides, subtract the same thing, multiply or divide both sides — all still work, and for the same reason: doing the same thing to both sides keeps the relationship true.

Add 3 to both sides of \(4 < 7\) and you get \(7 < 10\). Still true. Divide both sides of \(8 < 12\) by 4 and you get \(2 < 3\). Still true. The balance you built up in linear equations carries straight over: your goal is the same, to get \(x\) alone on one side. The only difference is the relationship in the middle is now a \(<\) or \(\ge\) instead of an \(=\).

The one rule everyone forgets: flip when you go negative

Now the exception — the single thing that makes inequalities different from equations. If you multiply or divide both sides by a negative number, you must reverse the direction of the inequality. The \(<\) becomes \(>\), the \(\ge\) becomes \(\le\).

This feels like a strange rule until you watch it happen. Take a fact nobody disputes:

Now multiply both sides by \(-1\). The left side becomes \(-3\), the right becomes \(-5\). Is \(-3 < -5\)? No — that is false. On the number line, \(-3\) sits to the right of \(-5\); it is the bigger number. To keep a true statement, the sign has to turn around:

That is the whole story. Multiplying by a negative reflects every number across zero, and reflecting flips the order: the number that was smaller lands on the bigger side. So whenever a negative multiplies or divides both sides, flip the sign to keep the statement honest. (Note that adding or subtracting a negative changes nothing about the direction — only multiplying or dividing by one does.)

Drawing the answer on a number line

A range is easiest to see. We picture the solution as a shaded ray sliding off along the number line, anchored at one boundary point. Two details do all the work:

• The endpoint. Use an open circle (hollow) for a strict \(<\) or \(>\), because the boundary itself is not included. Use a closed circle (filled) for \(\le\) or \(\ge\), because the boundary is a solution. This is exactly the "or equal to" underline showing up as ink.
• The direction. Shade toward the numbers that satisfy the inequality — left toward smaller values for \(<\) and \(\le\), right toward larger values for \(>\) and \(\ge\).

Here is the idea brought to life with \(x \ge -2\). Drag the boundary and switch the symbol with the buttons — watch the circle fill or hollow out, and the ray swing left or right.

When two conditions meet: compound inequalities

Sometimes a number must satisfy two conditions at once — "warmer than 10° but no hotter than 25°." We write that as a single squeezed expression: \[10 < t \le 25\] which reads "\(t\) is greater than 10 and at most 25." On the number line it is not a ray but a segment: an open circle at \(10\), a closed circle at \(25\), and the band between them shaded. You solve a compound inequality by doing the same operation to all three parts at once — and yes, if you multiply or divide the whole chain by a negative, every sign flips. We will explore these in depth later; for now, just recognise that a solution can be a bounded interval, not only an open-ended ray.

Practice

Try each one yourself, then reveal the full solution.