CAT Quant · Study & Practice

Modulus

AreaAlgebra DifficultyModerate CAT weightage1–2 questions (directly + inside inequalities, functions, coordinate geometry)

Modulus, or absolute value, looks deceptively simple — |x| is just "the distance of x from zero", always non-negative — but CAT loves it precisely because that one definition forces case-work, and case-work is where careless aspirants leak marks. The moment you write |x| you are really writing two expressions glued together: x when x is non-negative, and −x when x is negative. Master that piecewise view and a whole family of CAT questions opens up: modulus equations like |2x−3| = 7, sum-of-distance equations like |x−1| + |x−5| = 10, range questions built on the triangle inequality, and the V-shaped graphs that show up in maxima–minima and coordinate-geometry problems. The smart approach is almost never blind algebra; it is the number-line picture, where |x−a| is read as "distance from a". This chapter builds that fluency from the ground up — the definition and properties, solving equations by cases and by critical points, and reading and using the graph of y = |x−a| + b — each with worked CAT-style examples, the fastest method, and the traps that catch the unwary.

Topics

1 Absolute Value 4 worked · 5 practice · 8-Q test Start → 2 Modulus Equations 4 worked · 5 practice · 8-Q test Start → 3 Modulus Graphs 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Read |x − a| as "distance from a". Then |x − a| < c instantly means a − c < x < a + c.
  • For |x − a| + |x − b| = c: minimum is |a − b|. No solution if c < |a − b|; infinitely many if c = |a − b|; exactly two if c > |a − b|.
  • Solve |f(x)| = g(x) only after demanding g(x) ≥ 0, then split f(x) = ±g(x) and reject roots that break the sign condition.
  • y = |x − a| + b is a V with vertex (a, b), arms of slope ±1, and minimum value b at x = a — sketch it instead of doing algebra.
  • Count solutions of |x − a| + b = c by comparing c with b: two if c > b, one if c = b, none if c < b.
  • Use the triangle inequality |a + b| ≤ |a| + |b| to bound expressions fast in range/maximum questions.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Treating |x| = −x as always false — it is true for every x ≤ 0.
  • Forgetting to check the sign condition on the right side of |f(x)| = g(x), keeping a root where g(x) < 0.
  • Dropping a case: |ax + b| = c gives TWO equations, not one.
  • Solving |x − a| + |x − b| = c without spotting that c < |a − b| means no solution at all.
  • Mis-placing the vertex of y = |x − a| + b — the inside shift is +a (to the right), not −a.

📈 CAT exam insight & PYQ analysis

In recent CAT and XAT papers, pure modulus questions are rare as standalone items but recur woven into inequalities, functions, and coordinate geometry. The high-value recurring patterns are sum-of-distance equations |x − a| + |x − b| = c (often phrased as "number of integer/real solutions"), counting roots of |f(x)| = g(x), and minimum-value questions that reduce to the vertex or the |a − b| gap. Difficulty ranges Moderate to Hard; the trap is almost always a missed case or an unchecked sign. Aspirants who default to the number-line/graph picture finish these in seconds, while those who grind algebra blindly lose both time and accuracy.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

Definition of |x|?Tap to reveal
x if x ≥ 0, and −x if x < 0
|x| in terms of a square root?Tap to reveal
|x| = √(x²)
What does |x − a| represent geometrically?Tap to reveal
The distance between x and a on the number line
|x| = c (c ≥ 0) gives?Tap to reveal
x = c or x = −c
Solutions of |x| = c when c < 0?Tap to reveal
None — a modulus is never negative
|x| ≤ c (c ≥ 0) means?Tap to reveal
−c ≤ x ≤ c
|x| ≥ c means?Tap to reveal
x ≤ −c or x ≥ c
Triangle inequality?Tap to reveal
|a + b| ≤ |a| + |b|
Minimum of |x − a| + |x − b|?Tap to reveal
|a − b|, for any x between a and b
Vertex of y = |x − a| + b?Tap to reveal
(a, b)
Minimum value of y = |x − a| + b?Tap to reveal
b, attained at x = a
How many roots does |x − a| + b = c have if c > b?Tap to reveal
Two

📌 Quick revision

Modulus |x| is distance from 0: x if x ≥ 0, −x if x < 0, and |x| = √(x²). Read |x − a| as distance from a to turn inequalities into ranges instantly. Solve |ax + b| = c (c ≥ 0) by splitting into ±c; solve |f(x)| = g(x) only after g(x) ≥ 0, then reject bad roots. For |x − a| + |x − b| = c use critical points a and b: the minimum is |a − b|, so c < |a − b| gives no solution, c = |a − b| gives a whole interval, c > |a − b| gives two roots. The graph y = |x − a| + b is a V with vertex (a, b), slopes ±1, minimum b — use it to count solutions and find minima. Watch the traps: don’t drop a case, always check the sign condition, and place the vertex correctly.

Chapter test

📝 Modulus — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
📊 My CAT analytics
Track scores, accuracy by topic and progress over time.
View analytics →

🏆 Vidaara CAT success checklist

You have truly mastered Modulus when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (3 topics)3/3
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards