CAT 2025 Lesson : Inequalities - Concepts & Cheatsheet

1)

If then Example $a > b$, $k \in r$ $a + k > b + k$ $a - k > b - k$ $a - b > 0$ or $0 > b - a$ If $x > 5 + y$ ⇒ $x - y > 5$ ⇒ $-5 > y - x$ ⇒ $y - x < -5$ $a > b$, $k > 0$, $j < 0$ $ak > bk$ $aj < bj$ $2x - 10 > 2$ ⇒ $x - 5 > 1$ $a - b > -2$ ⇒ $\dfrac{a - b}{-2} < 1$ ⇒ $\dfrac{b - a}{2} < 1$ Note: $a > b$ cannot be written as $\dfrac{a}{b} > 1$ as we do not know if the variable $b$ is negative or positive.

2) Solving Linear Inequalities: Move the variable terms to one side and constants to the other side.

3) Solving Higher Order Inequalities:

(a) Move all terms to the LHS of the inequality, so that the RHS equals $0$.

(b) Factorise the expression on the LHS.

(c) Ensure the coefficient of $x$ in every factor is positive. Else, multiply the factor by $-1$ and change the sign.

(d) Equating each factor to $0$ will provide the deflection values of $x$. Plot these on a number line.

(e) The region to the right of the right-most deflection point will be positive. The signs of prior regions (between deflection points) will be alternating between positive and negative.

(f) Choose the ranges that satisfy the inequality.

4) $\mathopen|x\mathclose|$ = $\bm{x}$, if $x \ge 0$; and
           = $\bm{-x}$, if $x < 0$

5) Where $k$ is a constant,
if $\mathopen|x\mathclose| < k$, then $\bm{-k < x < k}$
if $\mathopen|x\mathclose| > k$, then $\bm{x < -k}$ or $\bm{x > k}$

6) Key properties of modulus involving $2$ variables are as follows.
1) $\mathopen| x + y \mathclose| \le \mathopen| x \mathclose| + \mathopen| y \mathclose|$
2) $\mathopen| x + y \mathclose| \ge \mathopen| x \mathclose| - \mathopen| y \mathclose|$
3) $\mathopen| x - y \mathclose| \ge \mathopen| x \mathclose| - \mathopen| y \mathclose|$

7) Where $a$ and $b$ are positive terms and $k$ is a constant,
If $a + b = k$, then $ab$ is maximum when $\bm{a = b}$ and $ab$ is minimum when $\bm{(a - b)}$ is greatest.
If $ab = k$, then $a + b$ is maximum when $(a - b)$ is greatest and $a + b$ is minimum when $\bm{a = b}$

8) Where $x + y + z$ = $k$ (a constant), the maximum value of $x^{a}y^{b }z^{c}$ is attained when $\dfrac{x}{a} = \dfrac{y}{b} = \dfrac{z}{c}$.