lesson
Updated 5 days ago · 3 views
If you walk 4 miles north to a park, you walked 4 miles. If you turn around and walk 4 miles south to get home, you still walked 4 miles—not negative 4.
Physical distance ignores direction, and mathematics captures this exact idea through absolute value.
How do we represent distance on a standard number line?
Distance From Zero
The absolute value of a real number is its distance from 0 on a number line.
In 1841, German mathematician Karl Weierstrass introduced two vertical bars ∣x∣ to denote this distance notation cleanly.
📊An interactive-style visual diagram of a horizontal number line from -6 to 6. Zero is marked clearly in the center with a prominent neutral dot. At -4, a vibrant blue dot is placed with an arched dimension line connecting -4 to 0, labeled 'Distance = 4 units' and '|-4| = 4'. At +4, another vibrant blue dot with a mirrored arched dimension line connecting +4 to 0, labeled 'Distance = 4 units' and '|4| = 4'. Clean modern styling, responsive down to 350px width, showing that both directions yield positive 4.
Because distance cannot be negative, ∣x∣≥0 for every real number x, and ∣0∣=0.
What happens when an expression has math operations both inside and outside the bars?
Evaluating Expressions
Absolute value bars act as grouping symbols, sitting at the very top level of order of operations along with parentheses.
You must completely simplify everything inside the vertical bars first before taking the absolute value.
📊A step-by-step visual evaluation card for the expression 3|2 - 7| + 4. Step 1 highlights the inside: '2 - 7 = -5', yielding 3|-5| + 4. Step 2 applies the absolute value: '|-5| becomes 5', yielding 3(5) + 4. Step 3 multiplies: '3 * 5 = 15', yielding 15 + 4. Step 4 adds: '15 + 4 = 19' with a green checkmark box. Numbered badge cards with subtle connecting flow arrows.