Inequalities in modulus

[2iim]

For what range of values of x will the following inequality hold true?

|x + 7| > |2x + 3|

[ag]

Should I be evaluating all four cases

case 1. when left hand side and right hand side are both positive

case 2: when left hand side is negative and right hand side is positive

case 3: reverse of case 2

case 4: when both sides are negative.

Do they ask such questions in CAT Anyways, I will certainly skip this one.

[2iim]

AG

you do not have to evaluate all four cases. though that is one way to solve the question and as you mentioned a time consuming one.

it is sufficient if you square both sides of the inequation if you have modulus on both sides.

if |P| > |Q|, then P^2 will be greater than Q^2.

So, if you find range of values that satisfy P^2 > Q^2, then you have found out range of values that satisfy |P| > |Q|

[ag]

so, |x + 7| > |2x + 3| can be solved as (x + 7)^2 > (2x + 3)^2.

pls correct me if I have got it wrong

if i proceed with this, i get the following answer
(x + 7)^2 > (2x + 3)^2

x^2 + 14x + 49 > 4x^2 + 12x + 9
i.e., 3x^2 - 2x - 40 < 0
=> 3x^2 - 12x + 10x - 40 < 0
=> 3x (x - 4) + 10(x - 4) < 0
=> (3x + 10)(x - 4) < 0

so range of values will be -10/3 < x < 4

Is this correct?

[2iim]

ag - your answer is absolutely correct.

whenever there is a mod on both sides of the inequality as I had mentioned earlier all that you need to do is to square the values on both sides of the inequality and find the range of values.