Question

Number of integral solutions of inequality $|x+3|>|2x-1|$ is

• A. $3$
• B. $4$
• C. $5$
• D. $2$

Answer 1

The correct option is B $4$

The inequality is

$|x+3|>|2x-1|$

$|x+3|-|2x-1|>0$

Case (1)

For $-\infty <x<-3$ , inequality is

$-(x+3)-\{-(2x-1\left)\right\}>0$

$-x-3+2x-1>0$

$x-4>0$

$x>4$

But we considered $-\infty <x<-3$

so there is no solution in this case.

Case (2)

For $-3\le x<\frac{1}{2}$

$(x+3)+(2x-1)>0$

$3x+2>0$

$x>-\frac{2}{3}$

so inequality is true for $-\frac{2}{3}<x<\frac{1}{2}$

so $x=0$ is an integral solution.

Case (3)

For $\frac{1}{2}\le x<\infty$

$(x+3)-(2x-1)>0$

$x+3-2x+1>0$

$-x+4>0$

$x<4$

so inequality is true for $\frac{1}{2}\le x<4$

$x=1,2,3$ are the integral solution.

So $4$ integral solutions exists for given inequality.