value of x satisfying the equality $| x^{2} + 8 x + 7 | = | x^{2} + 4 x + 4 | + | 4 x + 3 |$ for $x \in R$ are

(- 2, \infty)

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(\frac{3}{4}, \infty) \cup {- 2}

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[- \frac{3}{4}, \infty) \cup {- 2}

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(- \frac{4}{3}, \infty)

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Solution

The correct option is C $(- 2, \infty)$
As we know
$| a | + | b | = | a + b |$
$a b \geq 0$

$⟹ (x^{2} + 4 x + 4) (4 x + 3) \geq 0$

$⟹ (x + 2)^{2} (4 x + 3) \geq 0$

$⟹ x \in [- \frac{3}{4}, \infty) \cup$ { $- 2$ }

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