Inequalities, Modulus and Absolute Value Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Inequalities, Modulus and Absolute Value short notes for CAT: 31 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

The Only Modulus Definition You Must Know

Definition card

Summary

∣ x ∣ = {x, - x, x \geq 0 x < 0

x \geq 0

, then:

∣ x ∣ = x

x < 0

, then:

∣ x ∣ = - x

The expression inside the modulus decides the case, not the whole equation.

The CAT Method: Breakpoints First

Breakpoint method

This is the safest method for one-variable modulus equations.

Set each modulus expression equal to zero.

Split the number line at those breakpoints.

Remove modulus signs interval-wise.

Check whether each solution lies in its interval.

Pattern 1: Linear Equations With |x| and |y|

Two-variable sign cases

PYQ Pattern

Case 1:

x \geq 0, y \geq 0

Case 2:

x \geq 0, y < 0

Case 3:

x < 0, y \geq 0

Case 4:

x < 0, y < 0

The final check is compulsory: a solution found in a case must satisfy that case’s sign assumptions.

Shortcut Identity: |x + y| + |x - y|

A high-value identity

∣ x + y ∣ + ∣ x - y ∣ = 2 max (∣ x ∣, ∣ y ∣)

So:

∣ x + y ∣ + ∣ x - y ∣ = 2

means:

max (∣ x ∣, ∣ y ∣) = 1

This shortcut is especially useful for integer pair counting.

Inequalities, Modulus and Absolute Value: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Ability MCQ

Let $f (x) = x^{2} - 9$ . For what positive value of the constant $r$ does the equation $∣ f (x) ∣ = r$ have exactly three distinct real roots?

A.
3
B.
9
C.
0
D.
81

Correct Answer:

Step-by-Step Solution

Key idea: This is a three-roots modulus-quadratic question, recognisable because it asks for the specific constant $r$ that yields exactly three intersections with the absolute value graph.

Step 1: The graph of $y = ∣ f (x) ∣$ is formed by keeping the positive parts of $f (x)$ and reflecting the negative parts above the x-axis.

Step 2: The minimum value of $f (x) = x^{2} - 9$ is $- 9$ (occurring at $x = 0$ ). When reflected, this minimum becomes a local maximum (a "touching" point) at $y = - (- 9) = 9$ .

Step 3: A horizontal line $y = r$ will intersect the graph of $y = ∣ f (x) ∣$ in exactly three points only when it perfectly touches this reflected local maximum.

Step 4: Therefore, the line must be at the height of the reflected minimum:

$r = 9$

Answer: B.

Question 2 · Quantitative Ability MCQ

If $x < 0$ , which expression gives the value of $∣ x ∣$ ?

A.
-x
B.
x
C.
0
D.
x^2

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct modulus-definition question, recognisable because a sign condition is given and the task is to replace $∣ x ∣$ by the correct branch.

Step 1: Recall the definition of absolute value:

$∣ x ∣ = {x, - x, x \geq 0 x < 0$

Step 2: The question states $x < 0$ , so we must use the second branch.

Step 3: Therefore, for $x < 0$ , $∣ x ∣ = - x$ . This is positive because the negative of a negative number is positive.

Answer: A.

More short notes in this unit

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Inequalities, Modulus and Absolute Value Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Inequalities, Modulus and Absolute Value short notes for CAT: 31 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice quest

A question from this chapter

Question 1

Let f (x) = x^{2} - 9 . For what positive value of the constant r does the equation ∣ f (x) ∣ = r have exactly three distinct real roots?

Question 2

If x < 0, which expression gives the value of ∣ x ∣ ?

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