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AN1 · Interpret absolute value as distance on a number line

Learn to interpret absolute value as distance on a number line through clear examples and targeted practice.

Alberta Grade 11 Mathematics

Algebra and Number

Read distance before solving an equation

In Alberta Mathematics 20-1, absolute value describes distance, not a rule that simply erases a minus sign. A point at −2 is five units from 3, so the distance is ∣−2−3∣=5|-2-3|=5. Thinking in distances helps you interpret equations and decide whether an answer makes sense. The number line below is generated from the same endpoints as the worked example.

What you will learn

  • Interpret an absolute-value expression as a distance.
  • Calculate the distance between two signed numbers.
  • Check a pair of solutions to a distance equation.

Distance from zero

The distance from x to zero is |x|. Thus ∣−4∣=4|-4|=4 and ∣4∣=4|4|=4: both points are four units from zero. Distance is never negative, even if the coordinate is negative.
The sign of a coordinate gives its direction from zero. The absolute value gives only how far away it is. Keep these two meanings separate when reading a diagram or solving a problem.
|x|=\begin{cases}x,&x\geq0\\-x,&x<0\end{cases}

Distance between two positions

The distance between points a and b is ∣a−b∣|a-b|, which also equals ∣b−a∣|b-a|. Subtracting in either order gives opposite signs, but absolute value gives the same nonnegative distance.
For an equation such as ∣x−3∣=5|x-3|=5, ask which points lie five units from 3. One is to the right and the other is to the left; test both in the original expression.
d(a,b)=∣a−b∣d(a,b)=|a-b|

Worked example

Two points five units apart

Plot −2 and 3. Find the distance between them, then solve ∣x−3∣=5|x-3|=5.
Distance from −2 to 3
Distance from −2 to 3The filled endpoints mark −2 and 3; the highlighted segment is five units long.-5-4-3-2-10123456-23|3 − -2| = 5

The filled endpoints mark −2 and 3; the highlighted segment is five units long.

  1. Subtract the coordinates
    Distance ignores the direction of travel.
    ∣−2−3∣=∣−5∣=5|-2-3|=|-5|=5
  2. Move both ways from 3
    A point five units left of 3 is −2; a point five units right is 8.
    x=3-5=-2 or x=3+5=8
  3. Check both candidates
    The original absolute-value equation must equal five for both values.
    ∣−2−3∣=5,∣8−3∣=5|-2-3|=5, |8-3|=5
Answer: The distance is 5 units; the equation has solutions x=−2x=-2 and x=8x=8.
Check: Both values satisfy the original equation. The figure illustrates the distance from −2 to 3; 8 lies outside its displayed window.

Common mistakes and how to avoid them

Reporting −5 as a distance.
Correction: The subtraction may be negative, but the absolute value and the distance are nonnegative.
Giving only the point to the right.
Correction: Check both directions from the centre of a distance equation.

Lesson summary

  • Absolute value measures distance from zero.
  • The distance between two points is the absolute value of their difference.
  • A positive distance from one point usually gives two positions.

Check your understanding

Question 1

What is the distance from −4 to 2?
  1. 6 units
  2. −6 units
  3. 2 units
Show answer and explanation
6 units
∣−4−2∣=6|-4-2|=6.

Key terms

Absolute value
Distance from zero on a number line.
Distance
A nonnegative measure of separation between two positions.

Continue through Math 20-1

View the complete Math 20-1 Alberta Grade 11 Mathematics curriculum and lessons

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Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-1), expectation AN1. It is a study resource, not an official curriculum publication.

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