First slide

Theory of expressions

Question

The solution of the inequality $\frac{x + 7}{x - 5} + \frac{3 x + 1}{2} \geq 0$ is

Moderate

A

$[1, 3] \cup (5, \infty)$
B

$(1, 3) \cup (5, \infty)$
C

$(- \infty, 1) \cup (5, \infty)$
D

none of these

Solution

We have $\frac{x + 7}{x - 5} + \frac{3 x + 1}{2} \geq 0$
$\begin{array}{l} \Rightarrow \frac{2 x + 14 + 3 x^{2} - 14 x - 5}{2 (x - 5)} \geq 0 \\ \Rightarrow \frac{3 x^{2} - 12 x + 9}{2 (x - 5)} \geq 0 \\ \Rightarrow \frac{(x - 1) (x - 3)}{(x - 5)} \geq 0 \end{array}$
From the sign scheme, $x \in [1, 3] \cup (5, \infty)$

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