Question

If $|x-5|+|x+5|=10$, then

• A. the number of integral solutions is 10
• B. the number of integral solutions is 11
• C. the sum of all the integral solutions is 0
• D. all the solutions of the equation are rational numbers

Answer 1

Answer: (B), (C)

$|x-5|+|x+5|=10$
Case-I: $x\ge 5$, the equation becomes
$(x-5)+(x+5)=10$
$\Rightarrow 2x=10\Rightarrow x=5$ which satisfies the case, therefore accepted.
Case-II: $-5<x<5$ The above equation becomes
$-(x-5)+(x+5)=10\Rightarrow -x+5+x+5=10$
$\Rightarrow 10=10\text{ which is true. }$
$\text{ So, the solution is }x\in (-5,5)$
$\text{ Case-III: }x\le -5\text{, The above equation becomes }$
$-(x-5)-(x+5)=10\Rightarrow -x+5-x-5=10$
$\Rightarrow -2x=10$
$\Rightarrow x=-5\text{ which satisfies the above case so, accepted. }$
$\therefore \text{ final answer is }x\in [-5,5]\Rightarrow BC$