Absolute Value

Once you are comfortable working with positive and negative integers, a natural question arises: how far is a number from zero, regardless of which direction? That question is answered by absolute value — one of the most useful concepts in pre-algebra. Absolute value strips away the sign and tells you the magnitude of a number, which is exactly what you need when measuring distances, tolerances, temperature differences, and electrical signals.

What Is Absolute Value?

The absolute value of a number is its distance from zero on the number line. Distance is always zero or positive — you cannot walk a negative number of steps.

Notation: The absolute value of $x$ is written $|x|$.

Formal definition:

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

The second line may look strange — “negative $x$” — but remember that if $x$ is already negative, then $-x$ flips it to positive. For example, if $x = -7$, then $-x = -(-7) = 7$.

Example 1: Evaluating Basic Absolute Values

$|5| = 5 \qquad \text{(5 is 5 units from zero)}$

$|-5| = 5 \qquad \text{(-5 is also 5 units from zero)}$

$|0| = 0 \qquad \text{(zero is 0 units from itself)}$

Both $5$ and $-5$ are the same distance from zero — they are on opposite sides of zero but equally far away. Absolute value ignores the direction and reports only the distance.

Example 2: Evaluating More Absolute Values

$|-12| = 12$

$|3.7| = 3.7$

$\left|-\frac{2}{3}\right| = \frac{2}{3}$

Absolute Value with Operations

When absolute value appears in a larger expression, treat the absolute value bars like grouping symbols — evaluate the inside first, then take the absolute value.

Example 3: Absolute Value of an Expression

Evaluate $|3 - 10|$.

Step 1: Compute the expression inside the bars.

$3 - 10 = -7$

Step 2: Take the absolute value.

$|-7| = 7$

Example 4: Operations Outside the Bars

Evaluate $|{-4}| + |{-9}|$.

Evaluate each absolute value separately:

$|-4| = 4 \qquad |-9| = 9$

Then add:

$4 + 9 = 13$

Example 5: Subtraction of Absolute Values

Evaluate $|{-15}| - |7|$.

$|-15| = 15 \qquad |7| = 7$

$15 - 7 = 8$

Example 6: Absolute Value with Multiplication

Evaluate $3 \times |{-8}|$.

First, evaluate the absolute value: $|-8| = 8$.

Then multiply:

$3 \times 8 = 24$

Example 7: Nested Operations Inside Bars

Evaluate $|{-6 + 2}| + |{4 - 11}|$.

Inside the first bars: $-6 + 2 = -4$, so $|-4| = 4$.

Inside the second bars: $4 - 11 = -7$, so $|-7| = 7$.

$4 + 7 = 11$

Comparing Numbers Using Absolute Value

Absolute value is helpful for comparing how far numbers are from zero without worrying about sign.

Example 8: Which Is Further from Zero?

Which is further from zero: $-13$ or $9$?

$|-13| = 13 \qquad |9| = 9$

Since $13 > 9$, the number $-13$ is further from zero than $9$, even though $9$ is the larger number on the number line.

Ordering by Absolute Value

Sometimes you need to order numbers by their absolute value (magnitude) rather than by their actual value.

Order by absolute value: $-2,\ 7,\ -10,\ 3,\ -5$

Absolute values: $2,\ 7,\ 10,\ 3,\ 5$

Ordered by absolute value (least to greatest): $-2,\ 3,\ -5,\ 7,\ -10$

Absolute Value as Distance Between Two Numbers

The distance between any two numbers $a$ and $b$ on the number line is:

$\text{distance} = |a - b|$

This works regardless of which number you subtract from which, because $|a - b| = |b - a|$.

Example 9: Distance Between Two Points

Find the distance between $-8$ and $5$ on the number line.

$|-8 - 5| = |-13| = 13$

Or equivalently: $|5 - (-8)| = |5 + 8| = |13| = 13$.

The two points are $13$ units apart.

Example 10: Temperature Difference

The morning temperature was $-3\degree$F and the afternoon temperature was $14\degree$F. How much did the temperature change?

$|14 - (-3)| = |14 + 3| = |17| = 17\degree\text{F}$

The temperature changed by $17\degree$F.

Real-World Application: Electrician — Voltage Deviation

An electrician checking a residential outlet expects $120$ volts. The actual reading is $113$ volts. How far off is the reading?

$|113 - 120| = |-7| = 7 \text{ volts}$

The reading deviates by $7$ volts from the expected value. If the acceptable tolerance is $\pm 6$ volts ($114$ to $126$ V), this outlet is outside tolerance and needs investigation.

Absolute value gives the size of the deviation without worrying about whether the reading is above or below the target — both directions are equally concerning.

Real-World Application: HVAC — Temperature Tolerance

A thermostat is set to $72\degree$F with a $\pm 2\degree$F tolerance band. The system should kick on whenever the actual temperature $T$ satisfies:

$|T - 72| > 2$

If the room reaches $75\degree$F:

$|75 - 72| = |3| = 3$

Since $3 > 2$, the system activates. The absolute value captures both the “too hot” and “too cold” conditions in a single expression.

Common Mistakes to Avoid

1. Thinking absolute value can be negative. By definition, $|x| \geq 0$ for every number $x$. If your answer for an absolute value is negative, you made an error.
2. Applying absolute value before computing the inside. In $|3 - 10|$, you must compute $3 - 10 = -7$ first, then take $|-7| = 7$. A common mistake is computing $|3| - |10| = 3 - 10 = -7$, which is wrong.
3. Confusing $|a - b|$ with $|a| - |b|$. These are different: $|{-8} - 3| = |-11| = 11$, but $|-8| - |3| = 8 - 3 = 5$.
4. Forgetting that $|{-x}| = |x|$. Both $-x$ and $x$ are the same distance from zero. This is not “removing the negative sign” — it is measuring distance.
5. Not treating absolute value bars as grouping symbols. Always evaluate everything inside the bars before applying the absolute value.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Absolute value $|x|$ is the distance from $x$ to zero on the number line — it is always zero or positive
• For positive numbers and zero, $|x| = x$; for negative numbers, $|x| = -x$ (which flips the sign to positive)
• Treat absolute value bars like grouping symbols — evaluate the expression inside first, then take the absolute value
• $|a - b|$ gives the distance between $a$ and $b$ on the number line, regardless of order
• $|a - b|$ is not the same as $|a| - |b|$ — these are different operations
• Absolute value is used in real-world contexts like voltage tolerance, temperature bands, and measurement error
• In trades, absolute value answers the question “how far off is this reading?” without caring about the direction of the error

Return to Pre-Algebra for more topics in this section.