calendarBack
Quant

/

Algebra

/

Inequalities
ALL MODULES

CAT 2025 Lesson : Inequalities - Concepts & Cheatsheet

bookmarked
Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

8. Cheatsheet

1)
If then Example
a>ba > b,
krk \in r
a+k>b+ka + k > b + k
ak>bka - k > b - k
ab>0a - b > 0 or 0>ba0 > b - a
If x>5+yx > 5 + yxy>5x - y > 5
5>yx-5 > y - xyx<5y - x < -5
a>ba > b,
k>0k > 0, j<0j < 0
ak>bkak > bk
aj<bjaj < bj
2x10>22x - 10 > 2x5>1x - 5 > 1
ab>2a - b > -2ab2<1\dfrac{a - b}{-2} < 1ba2<1\dfrac{b - a}{2} < 1

Note: a>ba > b cannot be written as ab>1\dfrac{a}{b} > 1 as we do not know if the variable bb 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 00.

(b) Factorise the expression on the LHS.

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

(d) Equating each factor to
00 will provide the deflection values of xx. 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)
x\mathopen|x\mathclose| = x\bm{x}, if x0x \ge 0; and
           =
x\bm{-x}, if x<0x < 0

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

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

7) Where
aa and bb are positive terms and kk is a constant,
If
a+b=ka + b = k, then abab is maximum when a=b\bm{a = b} and abab is minimum when (ab)\bm{(a - b)} is greatest.
If
ab=kab = k, then a+ba + b is maximum when (ab)(a - b) is greatest and a+ba + b is minimum when a=b\bm{a = b}

8) Where
x+y+zx + y + z = kk (a constant), the maximum value of xaybzcx^{a}y^{b }z^{c} is attained when xa=yb=zc\dfrac{x}{a} = \dfrac{y}{b} = \dfrac{z}{c}.

Want to read the full content

Unlock this content & enjoy all the features of the platform

Subscribe Now arrow-right
videovideo-lock