Find the sum of all integral values of $x$ interval $[- 4, 100]$ satisfying $| 2 x \sqrt{(2 x - 1)^{2} |} = 1$

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Solution

$f (x) = ∣ ∣ ∣ 2 x \sqrt{{(2 x - 1)}^{2}} ∣ ∣ ∣ = 1 | 2 x | 2 x - 1 | | = 1 2 x - 1 < 0 x < \frac{1}{2} x \in [- 4, 0] - 2 x \times (- (2 x - 1)) = 1 4 x^{2} - 2 x = 1 {4 x}^{2} - 2 x - 1 = 0 x \in [0, \frac{1}{2}] 2 x (1 - 2 x) = 1 2 x - 4 x^{2} = 1 {4 x}^{2} - 2 x + 1 = 0 x > \frac{1}{2} {4 x}^{2} - 2 x = 1 {4 x}^{2} - 2 x - 1 = 0 sum = 0$
$∵$ No integral values possible.

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