Absolute value and inequalities on the number lineIB MYP Maths Extended: Revision notes

The absolute value of a number, written $|x|$, is its distance from zero on the number line. Distance is never negative, so $|x|\ge0$. For example $|7|=7$ and $|-7|=7$. Do any working inside the bars first: $|3-8|=|-5|=5$. The distance between two numbers $a$ and $b$ is $|a-b|$, so the distance between $-7$ and $3$ is $|3-(-7)|=10$.

An inequality compares two quantities that are not equal. $x<5$ means $x$ is less than 5. $x>5$ means greater than 5. $x\le5$ means less than or equal to 5, and $x\ge5$ means greater than or equal to 5. The symbols with a line underneath ($\le$, $\ge$) include the end value; the others ($<$, $>$) do not.

On a number line, draw a closed circle (filled in) at the end value when it is included ($\le$ or $\ge$). Draw an open circle when it is not included ($<$ or $>$). Then draw a line or arrow in the direction of the allowed values.

• $x\ge-2$: closed circle at $-2$, arrow to the right.
• $x<4$: open circle at $4$, arrow to the left.
• $-1<x\le4$: open circle at $-1$, closed circle at $4$, line between them.

Set notation describes all the numbers that satisfy a condition. $\{x:x>2\}$ is read "the set of all $x$ such that $x$ is greater than 2". To restrict the type of number, write the set after the $\in$ sign: $\mathbb{Z}$ means integers (whole numbers, including negatives) and $\mathbb{R}$ means all real numbers. Example: $\{x\in\mathbb{Z}:-2\le x<3\}=\{-2,-1,0,1,2\}$. Here the list has five members because $-2$ is included and $3$ is not.

Interval notation is a short way to write a range. Use a square bracket when the end is included and a round bracket when it is not.

• $-1<x\le4$ is $(-1,4]$.
• $2\le x\le9$ is $[2,9]$.
• $x\ge3$ is $[3,\infty)$, and $x<5$ is $(-\infty,5)$. Infinity is not a number you can reach, so $\infty$ always has a round bracket.

Because $|x|$ is a distance, $|x|<3$ means "$x$ is less than 3 units from zero", so $-3<x<3$. In general $|x-a|\le d$ means $x$ is within $d$ units of $a$, so $a-d\le x\le a+d$. For example $|x-27|\le2$ gives $25\le x\le29$, or $[25,29]$. By contrast $|x|>3$ means $x$ is more than 3 units away from zero, so $x<-3$ or $x>3$.