Solving Inequalities: Linear, Quadratic, and Modulus

Inequalities appear in every campus placement aptitude section, typically 2 to 5 questions per paper, split across linear, modulus, and quadratic types.

Inequality Basics and Notation

For any two real numbers a and b, exactly one of three things holds: a < b, a = b, or a > b. This is the trichotomy property. It is the foundation of every inequality chain. Unlike an equation with a single answer, an inequality has a solution set: the range of all real values that satisfy it.

The four symbols used in aptitude tests:

Symbol	Meaning
<	Strictly less than
>	Strictly greater than
≤	Less than or equal to
≥	Greater than or equal to

Note that ≤ and ≥ include the boundary value; < and > do not. In solution-set notation, x ≤ 5 includes x = 5, while x < 5 does not.

Core Properties: What Changes and What Doesn’t

Four properties govern inequality arithmetic. Three preserve the direction. One reverses it.

Addition and subtraction: adding or subtracting any real number from both sides leaves the direction unchanged.

• If a < b, then a + c < b + c for any real c.
• If a > b, then a - c > b - c for any real c.

Multiplication by a positive number: multiplying both sides by any positive number preserves the direction.

• If a < b and c > 0, then ac < bc.

Multiplication by a negative number (sign reverses): this is the rule that placement tests probe most often.

• If a < b and c < 0, then ac > bc.
• Concrete check: 2 < 5. Multiply both sides by -1: -2 > -5. Correct.

Transitive property: if a < b and b < c, then a < c. This allows chaining inequalities in multi-step problems.

The sign-reversal rule is the single most tested property in campus aptitude papers. Whenever a solution step requires dividing by a negative coefficient, the direction flips.

Solving Linear Inequalities

Linear inequalities rearrange exactly like linear equations with one exception: dividing by a negative coefficient reverses the direction.

Example 1: Basic Linear Inequality

• Given: 2x + 5 > 15
• Step 1: Subtract 5 from both sides: 2x > 10
• Step 2: Divide by 2 (positive, no flip): x > 5
• Solution set: all real numbers greater than 5.

Example 2: Inequality with a Negative Coefficient

• Given: -3x + 6 ≤ 0
• Step 1: Subtract 6 from both sides: -3x ≤ -6
• Step 2: Divide by -3 (negative, sign reverses): x ≥ 2
• Solution set: all real numbers greater than or equal to 2.

Compound inequalities apply each operation to all three parts simultaneously.

Example 3: Compound Inequality

• Given: 1 < 2x - 3 < 9
• Step 1: Add 3 throughout: 4 < 2x < 12
• Step 2: Divide by 2 (positive, no flip): 2 < x < 6
• Solution set: all real numbers strictly between 2 and 6.

Modulus (Absolute Value) Inequalities

The absolute value |a| is the distance of a from zero on the number line, always non-negative. Two properties that appear repeatedly in aptitude tests:

• |a| = 0 only when a = 0.
• Triangle inequality: |a + b| ≤ |a| + |b| for all real a, b.

Modulus inequalities split into two standard cases based on the direction:

• When |x - a| ≤ k: the solution is a single connected interval, a - k ≤ x ≤ a + k.
• When |x - a| ≥ k: the solution splits into two separate intervals, x ≤ a - k or x ≥ a + k.

The key distinction: a less-than modulus gives one interval; a greater-than modulus gives two.

Example 4: Modulus Less-Than or Equal

• Given: |x - 3| ≤ 4
• Expand: -4 ≤ x - 3 ≤ 4
• Add 3 throughout: -1 ≤ x ≤ 7
• Solution set: all x in the interval from -1 to 7, inclusive.

Example 5: Modulus Greater-Than

• Given: |2x + 1| > 5
• Case 1: 2x + 1 > 5 gives 2x > 4, so x > 2
• Case 2: 2x + 1 < -5 gives 2x < -6, so x < -3
• Solution set: x < -3 or x > 2.

Quadratic Inequalities and the Sign-Chart Method

Quadratic inequalities have the form f(x) > 0 or f(x) < 0 where f(x) is a quadratic expression. Completing the square is slow under time pressure. The sign-chart method is faster and works reliably for any factorable quadratic.

Five steps:

• Step 1: Move everything to one side so the inequality reads f(x) > 0 or f(x) < 0.
• Step 2: Factor f(x) into linear factors.
• Step 3: Find the roots, the values where f(x) = 0.
• Step 4: Mark the roots on a number line. They divide the line into intervals.
• Step 5: Test the sign of f(x) in one interval. For a quadratic with a positive leading coefficient, the sign alternates between intervals, starting positive at the far right. Select intervals where the sign matches the inequality direction.

Example 6: Sign-Chart on a Quadratic

• Given: x² - 5x + 6 > 0
• Step 1: Already in f(x) > 0 form.
• Step 2: Factor: (x - 2)(x - 3) > 0
• Step 3: Roots are x = 2 and x = 3.
• Step 4: Three intervals: x < 2, 2 < x < 3, x > 3.
• Step 5: Test each interval:
  • For x < 2 (try x = 0): (0 - 2)(0 - 3) = (-2)(-3) = 6, which is positive. Satisfies > 0.
  • For 2 < x < 3 (try x = 2.5): (0.5)(-0.5) = -0.25, which is negative. Does not satisfy.
  • For x > 3 (try x = 4): (2)(1) = 2, which is positive. Satisfies > 0.
• Solution set: x < 2 or x > 3.

For x² - 5x + 6 < 0, the solution is the complement: 2 < x < 3.

The AM-GM Inequality

A named result that appears in optimization questions. For any two non-negative real numbers a and b:

• Arithmetic Mean (AM) = (a + b) / 2
• Geometric Mean (GM) = the square root of ab
• The relationship: AM ≥ GM, meaning (a + b) / 2 ≥ √(ab)

Equality holds only when a = b. On placement tests, AM-GM gives the minimum sum given a fixed product.

Example 7: AM-GM Applied

• Given: ab = 16 and a, b > 0. Find the minimum value of a + b.
• By AM-GM: (a + b) / 2 ≥ √(ab) = √16 = 4
• Therefore a + b ≥ 8. Minimum is 8, achieved when a = b = 4.
• Answer: 8.

Placement Test Patterns and Timing

Campus aptitude tests cluster inequality questions into three recognisable patterns:

1. Linear rearrangement: a straightforward rearrangement with one sign-flip step hidden inside. The trap is a negative coefficient that students forget to flip.
2. Modulus range-finding: determine the valid range of x from an absolute-value constraint. The two-case split is required for any ≥ form; missing it halves the solution set.
3. Sign-chart quadratic: factor the quadratic and apply the sign-chart. Two to three steps, fast with practice.

Timing guidance: allocate 1 to 1.5 minutes per inequality question. If a question takes longer, check two things first: did you factor correctly, and did you flip the sign where required.

For how inequalities fit within the full aptitude section structure, the campus placement evaluation test guide covers section weights and timing across common campus formats. The Time and Work aptitude guide covers the rate-based calculation patterns that appear alongside inequalities in the same test section. Mu Sigma’s MuApt test weights analytical reasoning heavily; the Mu Sigma aptitude analysis maps exactly how pattern-recognition questions appear in analytics-company hiring. For a book list calibrated to campus test patterns, the placement preparation book guide covers the most widely used titles. IndiaBix’s inequalities section has practice sets that mirror campus paper difficulty. The TCS NQT careers page links to official preparation resources.

The sign-chart method works by dividing a problem space into intervals and testing each one. That same interval-by-interval reasoning transfers directly when probing how an AI model behaves across different input ranges. TinkerLLM’s hands-on exercises at tinkerllm.com, starting at ₹299, apply structured decomposition to real LLM workflows, a practical complement to the analytical reasoning built here.