Question

PLEASE FIND THE ANSWER OF THIS QUESTION ​

Answer 1

Given Question

Solve the inequality :-

$\rm \: \dfrac{5}{x - 1} \geqslant 2 \: \: (x \: \ne \: 1)$

$\red{\large\underline{\sf{Solution-}}}$

Given inequality is

$\rm \: \dfrac{5}{x - 1} \geqslant 2$

$\rm \: \dfrac{5}{x - 1} - 2 \geqslant 0$

$\rm \: \dfrac{5 - 2(x - 1)}{x - 1} \geqslant 0$

$\rm \: \dfrac{5 - 2x + 2}{x - 1} \geqslant 0$

$\rm \: \dfrac{7 - 2x}{x - 1} \geqslant 0$

$\rm \: \dfrac{ -(2x - 7)}{x - 1} \geqslant 0$

$\rm \: \dfrac{2x - 7}{x - 1} \leqslant 0$

can be further rewritten as

$\rm \: \dfrac{(2x - 7)(x - 1)}{(x - 1)^{2} } \leqslant 0$

$\bf\implies \:(2x - 7)(x - 1) \leqslant 0$

$\bf\implies \:1 < x \leqslant \dfrac{7}{2}$

$\bf\implies \:x \: \in \: \bigg(1, \: \dfrac{7}{2}\bigg]$

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ADDITIONAL INFORMATION

If a and b are two real positive numbers such that a < b then

$\boxed{\tt{ (x - a)(x - b) < 0 \: \: \rm\implies \:a < x < b \: }}$

$\boxed{\tt{ (x - a)(x - b) \leqslant 0 \: \: \rm\implies \:a \leqslant x \leqslant b \: }}$

$\boxed{\tt{ (x - a)(x - b) > 0 \: \: \rm\implies \:x < a \: \: or \: \: x > b \: }}$

$\boxed{\tt{ (x - a)(x - b) \geqslant 0 \: \: \rm\implies \:x \leqslant a \: \: or \: \: x \geqslant b \: }}$

$\boxed{\tt{ \frac{x - a}{x - b} < 0 \: \: \rm\implies \:a < x < b \: }}$

$\boxed{\tt{ \frac{x - a}{x - b} \leqslant 0 \: \: \rm\implies \:a \leqslant x < b \: }}$

$\boxed{\tt{ \frac{x - a}{x - b} > 0 \: \: \rm\implies \:x < a \: \: or \: \: x > b \: }}$

$\boxed{\tt{ \frac{x - a}{x - b} \geqslant 0 \: \: \rm\implies \:x \leqslant a \: \: or \: \: x > b \: }}$