1.3 Absolute values and inequalities

Absolute values and inequalities turn descriptions such as “within two units” or “at least five” into mathematics. Absolute value measures the size of a difference, regardless of its sign. An inequality describes a range of acceptable values instead of one exact answer. You will use both when comparing measurements, checking errors, or deciding whether a constraint is met.

This is foundational material. If you can compare negative numbers and solve a simple equation, you can start here. For a first pass, use Khan Academy's lessons on negative numbers and absolute value. The OpenStax section on inequalities adds longer examples and practice.

A basic idea: distance on a number line

The absolute value $|x|$ is the distance from $x$ to zero. So $|-3|=3$ and $|3|=3$: the two numbers lie on opposite sides of zero but equally far away. More generally, $|x-a|$ is the distance from $x$ to $a$.

The order on the line also tells you how to compare signed numbers: a number farther right is larger. Thus $-2>-5$, even though $5$ has the greater absolute value.

A quick tour of rules

Absolute values

The definition is

$$|x|=\begin{cases} x & \text{if }x\ge 0,\\ -x & \text{if }x<0. \end{cases}$$

Absolute value is never negative, and changing a sign does not change a distance: $|x|\ge0$ and $|-x|=|x|$. The distance between two numbers is $|a-b|=|b-a|$. Combining two changes cannot create a greater distance than adding their separate distances, so $|a+b|\le|a|+|b|$.

For $r\ge0$, two useful translations are

$$|x-a|\le r\quad\Longleftrightarrow\quad a-r\le x\le a+r,$$

$$|x-a|>r\quad\Longleftrightarrow\quad x<a-r\ \text{or}\ x>a+r.$$

“Within” gives one interval around $a$; “farther than” gives two regions outside it. A strict sign, $<$ or $>$, excludes the boundary. A sign with equality, $\le$ or $\ge$, includes it.

Inequalities and numerical comparisons

You can add or subtract the same number on both sides without changing the comparison. Multiplication depends on the sign:

Multiplying by a negative reverses the order: $2<5$, but $-2>-5$. The same rule applies when you divide by a negative. If the sign of a multiplier is unknown, find it or split into cases before multiplying.

Application 1: a measurement with tolerance

A sensor reports $47.6$ units. Its true reading $x$ can be at most $1.2$ units away from the reported value. What readings are possible?

1. Write the distance. The difference from $47.6$ is $|x-47.6|$. “At most” includes the limit, so $|x-47.6|\le1.2$.
2. Turn distance into a range. Subtract and add the tolerance: $47.6-1.2\le x\le47.6+1.2$.
3. Simplify and check. The range is $46.4\le x\le48.8$. Both endpoints are exactly $1.2$ units away; $49$ is $1.4$ away and does not qualify.

The sign of an error tells you whether the sensor read high or low. Its absolute value tells you how large the error was.

Application 2: an energy limit

A robot begins with $12$ energy units. Each move uses $3$ units, and it must keep at least $2$ units for its final action. What is the greatest whole number of moves it can make?

1. Write the remaining energy. After $m$ moves it has $12-3m$ units, so the requirement is $12-3m\ge2$.
2. Solve the inequality. Subtract $12$ to get $-3m\ge-10$. Divide by $-3$ and reverse the sign: $m\le10/3$.
3. Use the whole-number constraint. The robot can make at most $3$ moves. After $3$ moves it has $3$ units left; after $4$ it has $0$, which fails the requirement.

This question needs both the inequality rule and a comparison between a fractional bound and possible whole-number moves.