Solutions on a number line; adding and subtracting

A solution of an inequality is a value that makes it true. For \(x>-1\), the values \(0\), \(1.5\) and \(4\) are solutions, but \(-1\) and \(-2.5\) are not. Because there are infinitely many solutions, we show all of them on a number line.

An open circle is used with \(<\) and \(>\); a filled circle is used with \(\le\) and \(\ge\). The arrow shows that the solutions continue without end.

Why adding and subtracting keep the sign: adding the same number to both sides moves both numbers the same distance along the number line, so the one on the left stays on the left. For example, \(3<7\), \(3+5<7+5\) because \(8<12\), and \(3-10<7-10\) because \(-7<-3\).

Method

1. Add or subtract the same number on both sides to get the unknown on its own. The inequality sign does not change.
2. If the unknown ends up on the right, read the inequality from right to left: \(7\ge y\) means the same as \(y\le7\).
3. Draw the number line: a circle at the boundary value (open for \(<\) or \(>\), filled for \(\le\) or \(\ge\)) and a ray towards the solutions.
4. Check by substituting one value on the ray, and the boundary value, into the original inequality.

Example

Solve \(x+4<1\) and represent the solution on a number line.

\(\begin{aligned} x+4&<1 \\ x&<1-4 \\ x&<-3 \end{aligned}\)

Check: \(x=-4\) gives \(-4+4=0\), and \(0<1\) is true. The boundary \(x=-3\) gives \(1<1\), which is false, so the circle is open.

Check: the ray points the same way as the inequality sign only when the unknown is on the left. A common error is to draw \(5\ge y\) with the ray to the right; it means \(y\le5\), so the ray goes to the left.