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  4. Number of integral values of x satisfying the equation |x2-3 x-4|+|x2-5 x-6|=|2 x+2| is

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Q. Number of integral values of $x$ satisfying the equation $\left|x^2-3 x-4\right|+\left|x^2-5 x-6\right|=|2 x+2|$ is

Complex Numbers and Quadratic Equations

Solution:

$\left|x^2-3 x-4\right|+\left|x^2-5 x-6\right|=|2 x+2|$
image
$=\left|\left(x^2-3 x-4\right)-\left(x^2-5 x-6\right)\right| $
$|x|+|y|=|x-y| \text { is true if } x y \leq 0 $
$\Rightarrow \left(x^2-3 x-4\right)\left(x^2-5 x-6\right) \leq 0$
$\Rightarrow (x-4)(x+1)(x-6)(x+1) \leq 0 $
$ x \in\{-1\} \cup[4,6]$