the solution set of inequation |2x-3| |x+2| is - Maths - Logs Equations and Inequalities

Question

the solution set of inequation |2x-3| <|x+2| is

Manbar Singh · Accepted Answer

2x-3 < x+2⇒2x-3 - x+2 < 0We get x = -2; 32 as the critical points that divide the real line into 3 parts :-∞, -2; [2, 3/2); [3/2, ∞)Case 1 : When -∞ < x < -2In this case, 2x-3 = -2x-3and x+2 = -x+2Now, 2x-3 - x+2 < 0⇒-2x-3 + x+2 < 0⇒-x+5 < 0⇒x > 5But x < -2 So, the given inequation has no solution in this case Case 2 : When -2 ≤ x < 32In this case, 2x-3 = -2x-3and x+2 = x+2Now, 2x-3 - x+2 < 0⇒-2x-3 - x+2 < 0⇒-2x + 3-x - 2 < 0⇒3x > 1⇒x > 13Case 3 : When 32≤x<∞In this case, 2x-3 = 2x-3and x+2 = x+2Now, 2x-3 - x+2 < 0⇒2x-3 - x - 2 < 0⇒x - 5< 0⇒x < 5Hence, from 1, 2 and 3 the solution set is 13, 5