Solve the following inequalities and write the solution set using… - Vidyadip

Question

Solve the following inequalities and write the solution set using interval notation.$\frac{2 x}{x-4} \leq 5$

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Answer

$\frac{2 x}{x-4} \leq 5$
$\therefore \frac{2 x}{x-4}-5 \leq 0$
$\therefore \frac{2 x-5 x+20}{x-4} \leq 0$
$\therefore \frac{20-3 x}{x-4} \leq 0$
$\text { When } \frac{a}{b} \leq 0,$
$a \geq 0$ and $b < 0$ or $a \leq 0$ and $b > 0$
$\therefore $ either $20 – 3x \geq 0$ and $x – 4 < 0$ or $20 – 3x \leq 0$ and $x – 4 > 0$
Case I:$ 20 – 3x \geq 0$ and $x – 4 < 0$
Case I:$20-3 x \geq 0 \text { and } x-4<0$
$\therefore x \leq \frac{20}{3} \text { and } x<4$
$\therefore x<4 \ldots \ldots . \text { (l) }$
Case II: $20-3 x \leq 0$ and $x-4>0$
$\therefore x \geq \frac{20}{3} \text { and } x>4$
$\therefore x \geq \frac{20}{3} \ldots \ldots \text { (ii) }$
From (i) and (ii), we get
$x \in(-\infty, 4) \cup\left[\frac{20}{3}, \infty\right)$

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