Equations And Inequalities Question 8
Question: The number of integral roots of the equation $ | x-1 |+| x+2 |-| x-3 |=4 $ is
Options:
A) 0
B) 1
C) 2
D) 4
Show Answer
Answer:
Correct Answer: C
Solution:
- [c] $ | x-1 |+| x+2 |-| x-3 |=4, $ has three absolute value expressions, thus we divide the problem into four intervals:
(i) If $ x<-2 $ then
$ -(x-1)-(x+2)+(x-3)=4\Rightarrow x=-8 $
(ii) If $ -2\le x<1, $ then $ -(x-1)+(x+2)+(x-3)=4 $
$ \Rightarrow x=4\notin [-2,,1), $ hence rejected
(iii) If $ 1\le x<3, $ then
$ (x-1)+(x+2)+(x-3)=4\Rightarrow x=2 $
(iv) If $ x\ge 3, $ then
$ (x-1)+(x+2)-(x-3)=4\Rightarrow x=0\notin [3,\infty ), $
Hence rejected
$ \therefore $ Solution set is $ {-8,2} $ and both are integers
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