Note: The video for this module contains a summary of all the concepts covered in the Inequalities 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>b,
k∈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 ⇒ −2a−b<1 ⇒ 2b−a<1 |
Note: a>b cannot be written as ba>1 as we do not know if the variable b is negative or positive.
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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) ∣x∣ = x, if x≥0; and
= −x, if x<0
5) Where k is a constant,
if ∣x∣<k, then −k<x<k
if ∣x∣>k, then x<−k or x>k
6) Key properties of modulus involving 2 variables are as follows.
1) ∣x+y∣≤∣x∣+∣y∣
2) ∣x+y∣≥∣x∣−∣y∣
3) ∣x−y∣≥∣x∣−∣y∣
7) Where a and b are positive terms and k is a constant,
If a+b=k, then ab is maximum when a=b and ab is minimum when (a−b) is greatest.
If ab=k, then a+b is maximum when (a−b) is greatest and a+b is minimum when a=b
8) Where x+y+z = k (a constant), the maximum value of xaybzc is attained when ax=by=cz. 
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