The number of real solutions of the equation x | x + 5 | + 2 | x + 7 | - 2 = 0 is _______. [2024]

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Solution

Q.
The number of real solutions of the equation $x | x + 5 | + 2 | x + 7 | - 2 = 0$ is _______. [2024]

Ans.
(3)

Let $f (x) = x | x + 5 | + 2 | x + 7 | - 2$ and $f (x) = 0$

Case 1 : $x \geq - 5$

$\Rightarrow x^{2} + 5 x + 2 x + 12 = 0$

$\Rightarrow x^{2} + 7 x + 12 = 0 \Rightarrow x = - 3 or x = - 4$

Case 2 : $- 7 < x < - 5$

$\Rightarrow - x^{2} - 5 x + 2 x + 12 = 0$

$\Rightarrow - x^{2} - 3 x + 12 = 0 \Rightarrow x^{2} + 3 x - 12 = 0$

$\Rightarrow x = \frac{- 3 \pm \sqrt{57}}{2}$

$\Rightarrow x \neq \frac{- 3 + \sqrt{57}}{2}$ $[∵$ $- 7 < x < - 5]$

Case 3 : $x \leq - 7$

$\Rightarrow - x^{2} - 7 x - 16 = 0 \Rightarrow x^{2} + 7 x + 16 = 0$

No real solution

So Number of real solutions = 3

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