Modulus Functions and Modulus Equations
Read the modulus as distance from zero, sketch the V-shaped graph it produces, and solve equations and inequalities that contain one.
What a learner can do afterwards
- Sketch y equals the modulus of a linear expression and mark where the graph turns
- Solve a modulus equation by splitting into the two cases and rejecting solutions that fail the original
- Solve a modulus inequality and write the answer as an interval or a union of intervals
1 · Read
Read the bars as stretch from zero. The graph makes a V whose bend sits where the inner line hits zero. For y shaped by x minus 3, the bend lands at (3, 0).
Split each equation into twin cases, one per sign. For bars round 2x minus 4 set to 6, work 2x minus 4 as 6 and as minus 6. The roots are 5 and minus 1.
Each root must face the first equation. Some guests fail at the door, as in bars round x set to x plus 2, where 1 breaks the check and just minus 1 stays. Keep just the ones that pass.
Gaps use stretch sense too. Bars round x minus 1 under 4 means within 4 of 1, so minus 3 through 5. Past marks split outward with OR: bars round x minus 2 past 5 means x past 7 or x under minus 3.
Split by sign, test each guest in the first equation, and read gaps as near or far.
More to read
3.6 Absolute Value Functions
The graph of y = |x - 3| is a V with its turning point where the inside hits zero, at (3, 0). Right of 3 it climbs as y = x - 3; left of 3 it falls as y = 3 - x. Mark the turning point before plotting anything else.
Every modulus graph has this corner where the rule swaps. Find it by solving inside = 0, then draw each arm as a straight ray. The corner is the anchor for sketches, shifts, and inequalities alike.
2.7 Linear Inequalities and Absolute Value Inequalities
Modulus inequalities split by distance meaning: |x - 1| < 4 means within 4 of 1, so -3 < x < 5. Less than gives a single sandwich interval between two numbers.
Greater than splits outward instead: |x - 2| > 5 means x > 7 OR x < -3, two rays with OR between them. Writing AND there would claim the impossible overlap. Match the connective to the direction.
Absolute value & piecewise functions: FAQ
Solve modulus equations by splitting into two cases, one per sign of the inside. For |2x - 4| = 6, solve 2x - 4 = 6 and 2x - 4 = -6 to get x = 5 or x = -1. Then test each in the ORIGINAL equation.
Testing matters because some candidates fail, especially in trickier equations like |x| = x + 2 where x = 1 sneaks in from algebra but breaks the check. Reject anything that fails and keep only verified solutions.
Explore more
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.