Question

The number of integral values of x satisfying the linear inequality |x|+|x+1| < 5 is .

• A. 3
• B. 4
• C. 5
• D. 6

Answer 1

The correct option is B 4

Given that |x|+|x+1| < 5.
Now, $\left(x\right|=\left(\begin{array}{c}x,\mathrm{if}x\ge 0\\ -x\mathrm{if}x<0\end{array}\right(x+1|=(\begin{array}{c}x+1,\mathrm{if}x\ge -1\\ -x-1\mathrm{if}x<-1\end{array}\left(x\right|+\left(x+1\right|=(\begin{array}{c}x+x+1=2x+1,\mathrm{if}x\ge 0\\ -x+x+1=1,\mathrm{if}-1<x<0\\ -x-x-1=-2x-1,\mathrm{if}x\le -1\end{array}\mathrm{Now},(x|+(x+1|<5\mathrm{Case}I:\mathrm{If}x\ge 0(x|+(x+1|=2x+12x+1<5\Rightarrow x<2\Rightarrow x=0,1\mathrm{Case}\mathrm{II}:\mathrm{If}-1<x<0(x|+(x+1|=1\mathrm{which}\mathrm{is}\mathrm{less}\mathrm{than}5.\mathrm{No}\mathrm{integral}\mathrm{value}\mathrm{of}x\mathrm{in}\mathrm{this}\mathrm{interval}.\mathrm{Case}\mathrm{III}:\mathrm{If}x\le -1(x|+(x+1|=-2x-1-2x-1<5\Rightarrow -2x<6\Rightarrow x>-3\therefore x=-2,-1$
Therefore, the integral values of x satisfying the given linear inequation is -2, -1, 0, 1.
Number of values = 4