First slide

Theory of expressions

Question

If $x$ is real then $\frac{x^{2} - 2 x + 4}{x^{2} + 2 x + 4}$ takes values in the interval

Moderate

A

$[1 / 3, 3]$
B

$(1 / 3, 3)$
C

$(3, 3)$
D

$(- 1 / 3, 3)$

Solution

Let $y = \frac{x^{2} - 2 x + 4}{x^{2} + 2 x + 4} .$ Then,
$x^{2} (y - 1) + 2 x (y + 1) + 4 (y - 1) = 0$
$\Rightarrow 4 (y + 1)^{2} - 16 (y - 1)^{2} \geq 0 [∵ x is real]$
$\begin{array}{l} \Rightarrow (y + 1)^{2} - {2 (y - 1)}^{2} \geq 0 \\ \Rightarrow (3 - y) (3 y - 1) \geq 0 \Rightarrow 1 / 3 \leq y \leq 3 \end{array}$

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