Important Formulas for CAT Inequalities

If a > b and c > 0, then the following transformations preserve the inequality in the shown directions:

• a + c > b + c
• a - c > b - c
• ac > bc
• a / c > b / c

If a, b ≥ 0 and n is a positive integer, then

• a < b implies aⁿ < bⁿ
• 1 / aⁿ > 1 / bⁿ (for positive a, b)

If a < b and x > 0, then

If a > b and x > 0, then

Modular (Absolute-Value) Inequalities

Basic identities and inequalities for absolute value

• |x - y| = |y - x|
• |xy| = |x| · |y|
• |x + y| ≤ |x| + |y| (triangle inequality)
• |x + y| ≥ ||x| - |y||

These relations are useful to bound expressions involving absolute values and to convert absolute-value inequalities into simpler forms. Typical methods: isolate the absolute expression and split into two linear inequalities, or square both sides when signs allow.

Quadratic Inequalities

Sign analysis using factorisation

Given a quadratic expressed as a product of linear factors, (x - a)(x - b), the sign of the product depends on the position of x relative to the roots a and b. Assume a < b. Then

• (x - a)(x - b) > 0 ⇒ x < a or x > b
• (x - a)(x - b) < 0 ⇒ a < x < b

Method: find the real roots a and b, plot them on the number line, and test intervals to determine where the quadratic is positive or negative.

Example: Solve (x - 1)(x - 4) > 0.

First, identify the roots: x = 1 and x = 4.
Consider the intervals produced by these roots: (-∞, 1), (1, 4), (4, ∞).
For x in (-∞, 1), both factors (x - 1) and (x - 4) are negative, so their product is positive.
For x in (1, 4), (x - 1) is positive and (x - 4) is negative, so the product is negative.
For x in (4, ∞), both factors are positive, so the product is positive.
Therefore, the solution set is x < 1 or x > 4.

Means and the AM-GM-HM Inequalities

Relation between arithmetic, geometric and harmonic means

For any set of positive real numbers a₁, a₂, ..., a_n, the following ordering holds:

• AM ≥ GM ≥ HM

Explicitly,

• (a₁ + a₂ + ... + a_n) / n ≥ (a₁ · a₂ · ... · a_n)^1/n

When all numbers are equal, equality holds. The AM-GM inequality is frequently used to find maxima or minima under sum/product constraints.

If a and b are positive quantities, then

If a, b, c, d are positive quantities, then

From the previous relation follows

• a⁴ + b⁴ + c⁴ + d⁴ ≥ 4abcd

For n positive quantities a₁, a₂, ..., a_k and a natural number m, the following generalisation holds:

EduRev's Tip

• 
• For any positive integer n, 2 ≤
• a^mbⁿc^p……..will be greatest when
• If a > b and both are natural numbers, then
  ⇒ a^b < b^a {Except 3² > 2³ & 4² = 2⁴}
• (n!)² ≥ nⁿ
•  If the sum of two or more positive quantities is constant, their product is greatest when they are equal and if their product is constant then their sum is the least when the numbers are equal.
  ⇒ If x + y = k, then xy is greatest when x = y
  ⇒ If xy = k, then x + y is least when x = y

Common Applications and Problem Strategies

Use the following approaches when faced with inequality problems:

• Isolate terms and, where possible, reduce to a known inequality (AM-GM, Cauchy-Schwarz, rearrangement, etc.).
• For absolute-value inequalities, remove absolute signs by splitting into cases based on the sign of the inside expression.
• For polynomial inequalities, factorise and use a sign-chart on the number line to find intervals where the expression is positive or negative.
• When maxima or minima under constraints are required, consider equality cases of AM-GM or use Lagrange multipliers for multivariable continuous problems (if within scope).

Summary. The rules collected here cover elementary manipulations of inequalities, absolute-value relations, sign analysis for quadratics, standard mean inequalities and several useful formulae and tips. Keep the provided images as reference formulae and use AM-GM along with factorisation and sign charts as first-line tools when solving quantitative-aptitude inequality problems.