Finding Absolute Value of Positive and Negative Numbers

Find: $|8|$ and $|-8|$

For $|8|$: How far is 8 from zero?: Count from 0 to 8 on the number line = $|8| = 8$

For $|-8|$: How far is $-8$ from zero?: Count from $-8$ to 0 on the number line = $|-8| = 8$

Compare the results: Both 8 and $-8$ are 8 units from zero = $|8| = |-8| = 8$

Answer: $|8| = 8$ and $|-8| = 8$. Opposites have the same absolute value!

Comparing Absolute Values

Which is greater: $|-7|$ or $|4|$?

Find $|-7|$: $-7$ is 7 units from zero = $|-7| = 7$

Find $|4|$: $4$ is 4 units from zero = $|4| = 4$

Compare: $7 > 4$ = $|-7| > |4|$

Answer: $|-7| = 7$ is greater than $|4| = 4$

Absolute Value in Expressions

Simplify: $|-3| + |5| - |-2|$

Find each absolute value: $|-3| = 3$, $|5| = 5$, $|-2| = 2$ = All values found

Substitute into expression: $3 + 5 - 2$ = Expression simplified

Calculate: $3 + 5 = 8$, then $8 - 2 = 6$ = $6$

Answer: $|-3| + |5| - |-2| = 3 + 5 - 2 = 6$

Finding Distance Between Numbers

Find the distance between $-4$ and $3$ on the number line.

Use the distance formula: Distance $= |a - b|$ or $|b - a|$ = Formula ready

Calculate $3 - (-4)$: $3 - (-4) = 3 + 4 = 7$ = Difference is $7$

Take absolute value: $|7| = 7$ = Distance is 7 units

Verify by counting: From $-4$ to $3$: count $-4, -3, -2, -1, 0, 1, 2, 3$ (7 steps) = Confirmed!

Answer: The distance between $-4$ and $3$ is $7$ units.

Mistake: Thinking $|-5| = -5$

Why: Students may not apply the "distance" concept and just keep the negative sign.

Correct: Absolute value is always non-negative. $|-5| = 5$ because $-5$ is 5 units from zero.

Mistake: Confusing $-|5|$ with $|-5|$

Why: These look similar but mean different things!

Correct: $|-5| = 5$ (find absolute value of $-5$), but $-|5| = -5$ (take the negative of $|5|$).

Mistake: Thinking absolute value changes the number inside

Why: Students might think $|-3|$ means "remove the negative" for any expression.

Correct: Absolute value gives the distance from zero. It doesn't change the number; it measures how far it is from zero.

Measurement Error

Scientists and engineers use absolute value to measure how accurate their measurements are.

If the actual temperature is 20 degrees Celsius and your thermometer reads 18 degrees Celsius, the error is $|18 - 20| = |-2| = 2$ degrees.

Temperature Changes

When comparing temperature changes, we often care about the size of the change, not the direction.

If the temperature dropped from 15 degrees Celsius to 8 degrees Celsius, the change is $|8 - 15| = |-7| = 7$ degrees.

Q: Why is absolute value always positive?

A: Absolute value represents distance, and distance cannot be negative. You can't walk a negative number of steps! Whether you go left or right, you still cover a positive distance.

Q: What is $|0|$?

A: $|0| = 0$. Zero is zero units away from itself. It's the only number whose absolute value equals zero.

Q: Is $-|x|$ the same as $|-x|$?

A: No! $|-x| = |x|$ (same distance from zero), but $-|x|$ is the negative of the absolute value. For example: $|-5| = 5$, but $-|5| = -5$.