Integers & Absolute Value • Topic 3 of 3

Absolute Value & Applications

What is Absolute Value?

The absolute value of an integer is its distance from zero on the number line, regardless of direction. It is always non-negative (zero or positive).

Symbol: Absolute value is written as

\[ |a|\) (vertical bars on both sides)

Definition:

\[ |a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases} \]

Examples:

\(|5| = 5 \]

(distance from 0 to 5 is 5 units)

$|-3| = 3$ (distance from 0 to -3 is 3 units)

$|0| = 0$

Key Properties of Absolute Value:

Property	Rule	Example
Non-negativity	\(\	a\	\geq 0\)	\(\	-7\	= 7 \geq 0\)
Identity	\(\	a\	= 0\) if and only if $a = 0$	\(\	0\	= 0\)
Multiplicative	\(\	a \times b\	= \	a\	\times \	b\	\)	\(\	(-3) \times 4\	= \	-12\	= 12\) and \(\	-3\	\times \	4\	= 3 \times 4 = 12\)
Triangle inequality	\(\	a + b\	\leq \	a\	+ \	b\	\)	\(\	5 + (-3)\	= \	2\	= 2 \leq 5 + 3 = 8\)

Applications in Daily Life:

Temperature Changes: The absolute change in temperature ignores whether it increased or decreased
Bank Balances: The amount of debt (absolute value of negative balance)
Distance Traveled: Distance is always positive, direction doesn't matter
Sports Scores: Difference between scores (margin of victory)
Elevation: Depth below sea level expressed as positive number (absolute value)
Error Measurement: How far off a measurement is from the true value

Solved Examples

Worked Example

Example 1: Find the value of: $|-12| + |5| - |-3|$

Solution- Step 1: Find absolute value of each term: $|-12| = 12$ - Step 2: $|5| = 5$ - Step 3: $|-3| = 3$ - Step 4: Substitute: $12 + 5 - 3$ - Step 5: $12 + 5 = 17$, then $17 - 3 = 14$

Worked Example

Example 2: The temperature in a city was $-8^\circ$C in the morning. By afternoon, it rose to $7^\circ$C. What is the absolute change in temperature?

Solution- Step 1: Identify initial temperature = $-8$ - Step 2: Identify final temperature = $7$ - Step 3: Change = final - initial = $7 - (-8) = 7 + 8 = 15$ - Step 4: Absolute change = $|15| = 15$ - Step 5: The absolute value makes the change positive, representing the magnitude of change

Worked Example

Example 3 (Multi-step): A submarine is at a depth of 250 m below sea level. It ascends 120 m, then descends 45 m. What is its final position relative to sea level? What is the total distance traveled?

Solution- Step 1: Represent sea level as 0, below sea level as negative - Step 2: Starting position = $-250$ m - Step 3: Ascends 120 m: $-250 + 120 = -130$ m - Step 4: Descends 45 m: $-130 + (-45) = -175$ m - Step 5: Final position = $-175$ m (175 m below sea level) - Step 6: Total distance = ascends 120 m + descends 45 m = $120 + 45 = 165$ m - Step 7: OR using absolute values: $|120| + |-45| = 120 + 45 = 165$ m

Key Points

Absolute value of a number is its distance from zero, always non-negative
Symbol: $|a|$ means absolute value of $a$
$|a| = a$ if $a$ is positive or zero; $|a| = -a$ if $a$ is negative
Properties: $|a \times b| = |a| \times |b|$ and $|a + b| \leq |a| + |b|$
Distance between two points on number line = $|a - b|$
Real-life uses: temperature changes, distance, error measurement, debt amounts, elevation differences
Absolute value removes negative sign to show magnitude only
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.$|-5|=$

Explanation: Distance from $0$.

Q2.$|a|$ represents the ___ of $a$ from $0$.

Explanation: Distance.

Q3.$|0|=$

Explanation: $0$.

Q4.$|a|$ is always:

Explanation: Non-negative.

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