Quarry School

Absolute value measures distance

Explain it like I am five

Imagine your home at 0 on a straight road. You can walk six blocks right to 6 or six blocks left to −6. Your direction changes, but your trip is six blocks long either way. Absolute value asks for that length. The two upright marks in |x| are called absolute value bars. Read |x| aloud as the absolute value of x. They ask for the distance from x to zero on a number line, a road of numbered positions. A distance can be zero if you stay home. It cannot be negative. Finish every operation inside the bars before measuring that distance. Then do the operations outside them.

−8−7−6−5−4−3−2−1012−3−4lands on −7
Two leftward moves end at −7, which is seven units from zero.
Reminder
  • Order of operations. Grouping first, multiplication next, addition last: 2|3 − 5| + 1 = 2|−2| + 1 = 2 × 2 + 1 = 5.
  • A number beside bars. 3|x + 2| means 3 × |x + 2|, so at x = 0 it is 3 × |2| = 6.
Why it works. To the right of zero, the number already tells you the distance, so |x| = x when x ≥ 0. To the left, the number carries a minus sign for direction. Multiplying that number by −1 reverses its direction and gives its positive length, so |x| = −x when x < 0. For x = −6, −x means −(−6) = 6. The bars do not erase individual signs inside an unfinished calculation. They measure the result of the whole calculation.
Rule|x| = x when x ≥ 0; |x| = −x when x < 0. Nonnegative means zero or positive. Opposite means the same distance from zero on the other side: the opposite of −7 is 7. Always |x| ≥ 0, and |x| = 0 exactly when x = 0. A piecewise function chooses one instruction according to the input: use the first line for zero or positive inputs, the second for negative inputs.
The same idea, five ways
Say it

Say the absolute value of x.

Write it

Absolute value measures how far x is from zero, without its direction.

In math
  • |x|
  • |x| = x when x ≥ 0
  • |x| = −x when x < 0
  • |x| ≥ 0
Like

An address says which side of home; a measuring tape says how far.

See it
−6−4−20246−6lands on −6
The destination is −6, and its distance from zero is 6.
The same idea, other ways
As a walk

A sign says which side of zero you are on. Absolute value counts how many unit gaps separate the destination from zero.

−6−4−20246−6lands on −6
The destination is −6 and the distance is 6.
As a measuring tape

A measuring tape reads the length between two spots. Turning the tape around does not change a six-unit length: |−6| = |6| = 6.

|−6| = 6
|6| = 6
|0| = 0
Opposite directions can give the same distance.
As two instructions

A piecewise function selects an instruction using a condition on the input. For x ≥ 0, keep x. For x < 0, use −x. These instructions cover every real input and both give zero or a positive distance.

|x| = x, when x ≥ 0
|x| = −x, when x < 0
Choose the line whose condition fits your input.
With numbers

The column under −5 shows output 5 because −(−5) = 5. The column under 5 also shows 5. The column under 0 shows 0 because no movement separates zero from itself.

input xoutput |x|−55−22002255↓ evaluate: input given, read the output below it
The output row records distances, so every output is zero or positive.
.1Nonnegative input

On the right side of home, your position already says how far you walked. Zero stays zero.

  • For x ≥ 0, |x| = x.
  • Nonnegative means positive or zero.
0[0, ∞)
Zero and every number to its right use the keep-the-input instruction.
Worked exampleA positive position and home

Find |4| and |0|.

−1012345+4lands on 4
A walk from zero to 4 has length 4.
What it asks. Measure each input's distance from zero.
Plan. Both inputs are zero or positive, so keep each input.
  1. |4| = 4.4 is already four units to the right of zero.
  2. |0| = 0.Zero is no distance from itself.
Answer
  • |4| = 4.
  • |0| = 0.
Check Neither answer is negative, and only the input 0 gives distance 0.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: A distance must be greater than zero.
You can be at the starting point itself.
✓ Instead: |0| = 0, so absolute value is nonnegative, allowing zero.
Tips and tricks
  • The equals bar in x ≥ 0 includes zero in this instruction.
.2Negative input

On the left side of home, your position includes direction. Its opposite gives the length of the trip.

  • For x < 0, |x| = −x.
  • −x means the opposite of x.
−6−5−4−3−2−101+5lands on 0
The trip from −5 back to zero has length 5.
Worked exampleThe opposite of a negative

Find |−5| using the piecewise rule.

−6−5−4−3−2−101+5lands on 0
Five units return you from −5 to zero.
What it asks. Use the correct piecewise instruction to measure the distance of −5 from zero.
Plan. Test the sign of the input, then replace x in −x with (−5).
  1. x = −5 < 0, so use −x.The negative-input instruction applies to positions left of zero.
  2. −x = −(−5) = 5.Taking the opposite of the negative input −5 gives positive 5, the length from −5 to zero.
Answer
|−5| = 5.
Check Count the five unit gaps between −5 and 0.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: At x = −5, the formula |x| = −x gives −5.
Both minus signs must be kept in the substitution.
✓ Instead: −x = −(−5) = 5.
Tips and tricks
  • Use parentheses around a negative substituted number.
.3Multipliers and minus signs outside the bars

A sign or number outside the bars waits for the distance to be calculated. Think of measuring a length before doubling it or assigning its opposite. The measured length itself stays nonnegative, even when the completed expression becomes negative.

  • 3|x + 2| means 3 × |x + 2|.
  • −|u| is the opposite of the completed distance |u|; u here names the inside number.
  • Outside multiplication comes before outside addition.
3 − 5 = −2Take distance,multiply by 2, add 15inputoutput
The distance is found before the outside arithmetic.
Worked exampleA multiplier and an addition outside

Find 2|3 − 5| + 1.

−22|input| + 15inputoutput
A measured distance 2 produces the final output 5.
What it asks. Calculate a distance, double it, and then add 1.
Plan. Follow the grouping from inside outward.
  1. 3 − 5 = −2, so 2|3 − 5| + 1 = 2|−2| + 1.The inside subtraction must finish first.
  2. |−2| = 2, so the expression is 2 × 2 + 1.The distance from −2 to zero is 2.
  3. 2 × 2 + 1 = 4 + 1 = 5.Outside multiplication comes before addition.
Answer
2|3 − 5| + 1 = 5
Check The inside destination lies two units from zero. Doubling that length gives 4, and adding 1 gives 5.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: 2|3 − 5| + 1 = |6 − 10 + 1|.
The outside addition is not part of the measured inside, and the grouping has changed.
✓ Instead: 2|3 − 5| + 1 = 2 × 2 + 1 = 5.
Tips and tricks
  • Underline the entire inside before doing anything outside.
Strategy: step by step
  1. Calculate the entire expression inside the bars without changing its individual signs.
  2. If the result is zero or positive, keep it.
  3. If it is negative, use its opposite, the positive number the same distance from zero.
  4. Finish outside operations afterward. A number beside the bars multiplies: 3|x + 2| means 3 × |x + 2|.
Strategy
Evaluate an expression with absolute value bars
1
Is the inside result negative?
YesTake its opposite: |−7| = 7.
NoKeep it: |4| = 4 and |0| = 0.
↓
2
Is there an operation outside the bars?
YesDo it after finding the distance. −|−7| = −7.
NoThe distance is the answer.
  1. Find the one number inside the bars.
  2. Use its distance from zero.
  3. Complete outside multiplication, then outside addition or subtraction.
Worked exampleCalculate first, measure second

Find |−3 − 4| and then −|−3 − 4|.

−8−7−6−5−4−3−2−1012−3−4lands on −7
The inside finishes at −7, seven units from zero.
What it asks. Find the value of the first expression, then find its opposite in the second expression.
Plan. Keep the subtraction inside unchanged. Calculate its result, measure the distance, and apply the outside minus sign last.
  1. −3 − 4 = −7.Starting three blocks left and moving four more left puts you at −7.
  2. |−3 − 4| = |−7| = 7.The location −7 is seven blocks from zero.
  3. −|−3 − 4| = −7.The minus sign outside the bars acts after the distance has been found.
Answer
  • |−3 − 4| = 7.
  • −|−3 − 4| = −7.
Check Walking from −7 to 0 takes seven unit steps. The second expression takes the opposite of that length, so its value is −7.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: |−3 + 5| = 8 after making both numbers positive.
The bars measure the completed sum. −3 + 5 = 2, so changing −3 to 3 changes the sum.
✓ Instead: |−3 + 5| = |2| = 2.
✗ Not this: −x must be negative.
The minus sign asks for the opposite of x. If x is negative, its opposite is positive.
✓ Instead: For x = −6, −x = −(−6) = 6.
Tips and tricks
  • Inside first, distance second, outside last.
  • For a piecewise function, test the input against the conditions and use only the instruction it satisfies.
  • A distance has size, while its original sign tells direction. Zero is an allowed distance.
Trap. Making individual numbers positive before finishing the inside changes the problem. In |−3 + 5|, the wrong calculation |3 + 5| gives 8; the correct inside is 2, so the answer is 2. In |2 − 9|, changing subtraction to addition gives 11; the correct inside is −7, so the answer is 7.