Absolute Value

Study Sheet

Absolute Value

Distance from zero

Distance Is Never Negative

Concept
The Big Idea

∣x∣|x| is how far xx is from zero, ignoring direction. Distance cannot be negative, so the answer is never negative.

Absolute value is distance from zero, and distance is never negative: ∣−7∣=7|{-7}| = 7 and ∣7∣=7|7| = 7. It ignores direction and keeps size. The distance between two numbers on the line is the absolute value of their difference, ∣a−b∣|a - b|, in either order.

Example
Why ∣−7∣=7|-7| = 7
−8−7−6−5−4−3−2−1012345678−7777
−7 and 7 are both 7 steps from zero.

−7-7 sits seven steps from zero, so its absolute value is 77 — the same distance as +7+7.

Example
Zero and decimals

∣0∣=0|0| = 0, because 00 is no distance from itself. ∣−2.5∣=2.5|-2.5| = 2.5: absolute value works for decimals and fractions too.

Side note
Where it goes wrong

Thinking ∣x∣|x| "makes the sign positive" and so writing ∣x∣=x|x| = x always. For x=−7x = -7 that is false; ∣x∣=−x|x| = -x when xx is negative, because −(−7)=7-(-7) = 7.

Try it
Try it: four absolute values

Find ∣−15∣|-15|, ∣9∣|9|, ∣0∣|0| and ∣−34∣|-\tfrac34|.

Show the answer

15\mathbf{15}, 9\mathbf{9}, 0\mathbf{0} and 34\mathbf{\tfrac34}: each is a distance from zero, so none is negative.

In Plain Terms & More Examples

Concept
In plain terms

Absolute value is distance from zero, so it is never negative — just drop the sign.

Example
Finding ∣−12∣|-12|

It is 1212 steps from zero, so ∣−12∣=12|-12| = 12.

Example
Finding ∣8∣|8|

It is already positive: ∣8∣=8|8| = 8.

Side note
Where it goes wrong

Moving a minus sign inside or outside. ∣−5∣=5|-5| = 5, but −∣5∣=−5-|5| = -5: a minus sign outside is applied after taking the absolute value. So −∣−5∣=−5-|-5| = -5 as well.

Try it
Try it: which is bigger?

Which is bigger: ∣−9∣|-9| or ∣6∣|6|? And which is bigger: −∣4∣-|4| or ∣−4∣|-4|?

Show the answer

∣−9∣=9|-9| = 9 and ∣6∣=6|6| = 6, so ∣−9∣\mathbf{|-9|} is bigger. −∣4∣=−4-|4| = -4 and ∣−4∣=4|-4| = 4, so ∣−4∣\mathbf{|-4|} is bigger.

Distance Between Numbers

Concept
Subtract, then take the size
−5−4−3−2−101234−43|3 − (−4)| = 7
4 steps from −4 to 0, then 3 more to 3: 7 in all.

The distance between aa and bb on the number line is ∣a−b∣|a - b|. Order does not matter: ∣3−(−4)∣=∣−4−3∣=7|3 - (-4)| = |-4 - 3| = 7.

Example
Ordering
−4−3−2−101234567−30|2||−6|
Once the absolute values are worked out, read the values from left to right.

Sort ∣−6∣,−3,∣2∣,0|{-6}|, -3, |2|, 0: the values are 6,−3,2,06, -3, 2, 0, so in order: −3,0,∣2∣,∣−6∣-3, 0, |2|, |{-6}|.

Example
Distance from −6-6 to −2-2
−7−6−5−4−3−2−101−6−24
Both on the negative side: count the steps between them.

∣−6−(−2)∣=∣−6+2∣=∣−4∣=4|-6 - (-2)| = |-6 + 2| = |-4| = 4. On the line, −6-6 and −2-2 are 44 steps apart.

Side note
Where it goes wrong

Subtracting the absolute values. The distance between −6-6 and 44 is not ∣−6∣−∣4∣=2|-6| - |4| = 2. The numbers are on opposite sides of 00, so their distances from zero add: 6+4=10=∣−6−4∣6 + 4 = 10 = |-6 - 4|.

Try it
Try it: distance

How far apart are −7-7 and 55? And which is greater, −∣−8∣-|{-8}| or −3-3?

Show the answer

∣−7−5∣=12|{-7} - 5| = \mathbf{12}. −∣−8∣=−8-|{-8}| = -8, which is less than −3-3, so −3\mathbf{-3} is greater.

Word Problems

Concept
How far and how much

Absolute value answers questions about size without direction: how far below sea level, how much is owed, how big a change was. Positions can be negative; distances and amounts cannot.

A debt is recorded as a negative balance, but the amount owed is a positive number of dollars: the absolute value of the balance. The same split, signed position against unsigned size, appears in temperatures, elevations and scores.

Example
Elevation
−100001000200030004000peak: 3366 mDeath Valley: −86 m
Drawn to scale, with 0 at sea level: the valley floor sits just below it.

Death Valley is at −86-86 m and a nearby peak at 33663366 m. Their elevation difference is ∣3366−(−86)∣=3452|3366 - (-86)| = 3452 m.

Example
Temperatures

At dawn it is −5∘-5^\circC and by noon 8∘8^\circC. The rise is ∣8−(−5)∣=∣13∣=13|8 - (-5)| = |13| = 13 degrees: 55 degrees up to zero and 88 more.

Side note
Where it goes wrong

Confusing "less" with "less owed". A balance of −$45-\$45 is less than a balance of −$30-\$30, but it means owing more: ∣−45∣=45>30=∣−30∣|-45| = 45 > 30 = |-30|. Compare positions on the number line; compare amounts with absolute value.

Try it
Try it: debt

Kai's account is at −$45-\$45. How much must he deposit to reach $0\$0? Express it with absolute value.

Show the answer

∣−45∣=$45|{-45}| = \mathbf{\$45}.

Comparing with Absolute Value

Concept
Greater, or farther?

"Which number is greater?" and "which number is farther from zero?" are different questions. For negative numbers they have opposite answers.

Greater means further right on the number line. Farther from zero means a larger absolute value. For positive numbers the two agree; for negatives they disagree, because moving left makes a number smaller but takes it farther from zero.

Example
Comparing −8-8 and −3-3
−9−8−7−6−5−4−3−2−101−8−38 from zero
Smaller, yet farther from zero.

−8<−3-8 < -3, because −8-8 is further left. But ∣−8∣=8>3=∣−3∣|-8| = 8 > 3 = |-3|: −8-8 is farther from zero.

Example
Solving ∣x∣=4|x| = 4
−5−4−3−2−1012345−4444
Two points, one on each side, 4 from zero.

Which numbers are 44 from zero? Two of them: x=4x = 4 and x=−4x = -4.

Example
Integers with ∣x∣<3|x| < 3

The integers less than 33 away from zero are −2,−1,0,1,2-2, -1, 0, 1, 2: five of them.

Side note
Where it goes wrong

Giving only one answer to ∣x∣=4|x| = 4. Both 44 and −4-4 are 44 units from zero, so both are solutions.

Try it
Try it: close to zero

List the integers with ∣x∣≤2|x| \le 2. Then find xx if ∣x∣=6|x| = 6 and x<0x < 0.

Show the answer

−2,−1,0,1,2\mathbf{-2, -1, 0, 1, 2}. ∣x∣=6|x| = 6 means x=6x = 6 or x=−6x = -6; the negative one is −6\mathbf{-6}.

Why It Works

By definition ∣x∣=x|x| = x for x≥0x \ge 0 and ∣x∣=−x|x| = -x for x<0x < 0. Both branches are non-negative, so ∣x∣≥0|x| \ge 0 always, and ∣x∣=0|x| = 0 only for x=0x = 0. The distance between aa and bb is ∣a−b∣|a - b|: if a≥ba \ge b the distance is a−ba - b; if a<ba < b it is b−a=−(a−b)b - a = -(a - b) — exactly the two branches of the definition.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.