Solving inequalities

Last time

Before we start: last time, in Linear programming and shaded regions, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

More than, less than

A lift is safe for at most eight people. A bus ticket is half price if you are under sixteen. Real rules are rarely equals: they are limits. An inequality describes every value that satisfies the limit. Today you solve them like equations, with one famous trap, and show the answers on a number line.

The four symbols

Less than, with a pointed end towards x, says x is smaller. Greater than says x is bigger. With a line underneath, the number itself is allowed. Less than or equal to, and greater than or equal to. Without the line, the number is not allowed: it is strict. A double inequality, minus three less than x, less than or equal to four, gives a range of values.

On a number line

On a number line, draw an open circle for a strict symbol, and a closed, filled circle when the end is included. For minus four less than x, less than or equal to three, put an open circle at minus four and a closed circle at three. Then join them with a line, because every value between them is allowed.

Drawing −4 < x ⩽ 3

Here is the number line for minus four less than x, less than or equal to three. The circle at minus four is open, because minus four is not included. The circle at three is closed, because three is included. The line between them shows all the values.