Inequalities and Inequations : CAT Quantitative Ability

‘Inequalities and Inequations’ is considered to be one of the more important topics from the CAT point of view. This fact seems to be true simply because this topic is very malleable from question-setting point of view, that is inequalities can be clubbed with many other Algebra topics, and hence finds favour with the CAT Examination Paper Setters.

In the topic on ‘Numbers’ we introduced the idea of order among the real numbers. If we pick two real numbers — ‘a and b’, then they are equal or one is greater than the other.

• If a number a is greater than b, write a >b or b<a.Although we may understand this intuitively, we need a solid definition. Assuming that we understand the idea of positive and negative number, we define a>b if and only if (a – b) > 0.
• At times we use the symbol “≥, ≤ ”. These denote the relation “less than or equal to” and “greater than or equal to”.For example all the real numbers 1, 2, 3, 4 are greater than or equal to 1.
• Note that a≥ b if and only if (a – b) ≥  0, i.e. if and only if (a – b) is non-negative. 6 ≥ 5, since 6 – 5 = 1 ≥  0
• Note that a>b, a = b or a<b. No two of these can hold simultaneously.

Manipulating Inequalities

• The rules are set out below. In these rules a, b and c are all real numbers, unless stated otherwise.

(i)  If a<b, then a + c<b + c
(ii)  If a<b and c> 0, then ac<bc
(iii) If a<b and c< 0, then ac>bc
(iv) If a<b, then –a> –b

The reasons for the above will become clear if we examine few examples.

Examples:

(i) Let a = 3, b = 4, c = -2
    Now a = 3 < 4 = b, a + c = 3 – 2 = 1, b + c = 4 – 2 = 2
    a + c = 1 < 2 = b + c
Alternatively, (a + c) – (b + c) = (a – b) + (c – c) = a – b
Now, a<b gives a – b< 0 which gives us (a + c) – (b + c) < 0
a + c<b + c

(ii) Let a = 5, b = 7 and c = 2 > 0
a = 5 < 7 = b, so ac = 10 < 14 = bc

(iii) Let a = 9, b = 11, c = - 4
Now, a = 9 < 11 = b
ac = -36 > -44 = bc

(iv) Let a = 3, b = 5
a = 3 < 5 = b and –a = -3 > -5 = -b

• If in the above rules we replace ‘<, >’ by ‘ ,, ≤ ≥  ’, then it does not affect the rules in any way.

Example : If 2 ≤ 3, then 2 + 4 = 6 ≤7 = 3 + 4

More Results

• Look at the inequality 3 < 5 – 1. It follows that 3 + 1 < 5. So, if a – c>b then a>b + c

• If a>b and b>c, then a>c

Example :
Let a = 9, b = 2, c = -4
Now a = 9 > 2 = b and b = 2 > -4 = c
It is clear that a = 9 > -4 = c

a>b and c>d, then a + c>b + d
Let a = -5, b = -7, c = 16, d = 14
a = -5 > -7 = b, c = 16 > 14 = d
a + c = -5 + 16 = 11, b + d = -7 + 14 = 7
a + c = 11 > 7 = b + d
Inequality as a concept is used in many entrance examinations in Quantitative Aptitude Sections.  The beauty of this section is that questions on Inequalities are never asked directly but the concept is well applicable in many questions.