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CDS Mathematics · Inequalities

Signs, Powers and Comparisons

Many inequality questions need only signs: an even power is never negative, an odd power keeps the sign, and x + 1/x ≥ 2 for every positive x.

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Why this matters

Eight PYQs, two of them HARD, mostly data-sufficiency items. Read each statement for what it forces about signs — x⁸y⁹ < 0 forces y < 0 but says nothing about x — and test borderline values (a negative n, r between 1 and 1.26) before calling a statement sufficient.

Concept 1 of 2: What a statement forces about signs

An even power hides the sign of its base; an odd power shows it. So a sign condition on a product of powers tells you only about the bases raised to odd powers.

Definition

  • x2k≥0x^{2k} \ge 0 always; x2k+1x^{2k+1} has the sign of xx.
  • mn>1\dfrac mn > 1 gives m>nm > n only when n>0n > 0; for n<0n < 0 it gives m<nm < n.
  • ∣r∣<1|r| < 1 makes ∣rn∣<1|r^n| < 1 for every natural nn; rr just above 11 makes rnr^n grow.
  • For x<0x < 0: 2x<x<−x2x < x < -x, and kx<0kx < 0 for every natural kk.
  • To show a statement is NOT sufficient, find two cases that fit it with different answers.

Odd and even powers

x2k≥0,sign⁡ ⁣(x2k+1)=sign⁡(x)x^{2k} \ge 0, \qquad \operatorname{sign}\!\left(x^{2k+1}\right) = \operatorname{sign}(x)

Worked example

Is ab<0ab < 0? (I) a4b5>0a^4b^5 > 0. (II) a3b6<0a^3b^6 < 0.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q30Moderate

Example 1 · Inequalities · Signs, Powers and Comparisons

Consider the question and two statements given below : Let x and y be two real numbers. Question : Is xy>0xy > 0 ? Statement-1 : x8y9<0x^8 y^9 < 0. Statement-2 : x9y10<0x^9 y^{10} < 0. Which one of the following is correct in respect of the question and the statements ?

Dividing by an unknown sign

mn>1\dfrac mn > 1 does not mean m>nm > n unless nn is positive. With n=−2n = -2 and m=−3m = -3 the ratio exceeds 11 but m<nm < n.
Drill 5 more on what a statement forces about signs

Concept 2 of 2: x + 1/x and comparing expressions

For positive numbers, the arithmetic mean is never below the geometric mean. Applied to xx and 1x\tfrac1x, whose product is 11, that gives x+1x≥2x + \tfrac1x \ge 2.

Definition

  • x>0⇒x+1x≥2x > 0 \Rightarrow x + \dfrac1x \ge 2, with equality only at x=1x = 1.
  • To compare two expressions, look at their difference (or ratio) and factorise it.
  • (a3+b3)(a+b)−(a2+b2)2=ab(a−b)2≥0(a^3 + b^3)(a + b) - (a^2 + b^2)^2 = ab(a - b)^2 \ge 0 for positive a,ba, b.
  • A statement saying 'only when a>ba > b' is false if it also holds when b>ab > a.

AM–GM for x and 1/x

x+1x≥2(x>0)x + \dfrac{1}{x} \ge 2 \quad (x > 0)

Worked example

For positive xx, is (x+1x)3>7\left(x + \dfrac1x\right)^3 > 7 always?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (II) 2025 — Elementary Mathematics · Q2Moderate

Example 2 · Inequalities · Signs, Powers and Comparisons

Consider the following in respect of a positive real number xx : I. x+1x>1x+\frac{1}{x}>1 II. (x+1x)2>2\left(x+\frac{1}{x}\right)^2>2 III. (x+1x)4>9\left(x+\frac{1}{x}\right)^4>9 Which of the above are correct?

'Only when' is a claim too

An inequality that holds for every pair of unequal positive numbers does not hold 'only when a>ba > b'. The word 'only' makes the statement false.
Drill 1 more on x + 1/x and comparing expressions

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (2)

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