Chapter 1.9, Problem 67E

Solution

Simplify the inequality $6x^2-5x+2<-3x^2+x$ by any method.

The solution for the inequality is doesn’t exist.

Given:

The inequality is $6x^2-5x+2<-3x^2+x$ .

Concept used:

Square of a number can never be negative.

Calculation:

Simplify the inequality $6x^2-5x+2<-3x^2+x$ algebraically.

  $\begin{array}{c}6x^2-5x+2<-3x^2+x\mathrm{......}\left(\text{Given}\right)\\ \text{or, }9x^2-6x<-2\\ x^2-\frac{2}{3}x<-\frac{2}{9}\mathrm{......}\left(\text{Divide by 9 both sides}\right)\\ x^2-\frac{2}{3}x+{\left(\frac{1}{3}\right)}^2<-\frac{2}{9}+{\left(\frac{1}{3}\right)}^2\mathrm{......}\left(\text{Add }{\left(\frac{1}{3}\right)}^2\text{both sides}\right)\\ {\left(x-\frac{1}{3}\right)}^2<-\frac{1}{9}\mathrm{......}\left(\text{which is impossible}\right)\\ x\notin R\mathrm{......}\left(\text{where }R\text{is any real number}\right)\end{array}$

Hence the solution for the inequality doesn’t exist.