Mechanics Q (1 Viewer)

krypticlemonjuice · Jul 12, 2023

Why did they times (x-2) on the numerator and denominator in this q?

tywebb · Jul 13, 2023

krypticlemonjuice said:
Why did they times (x-2) on the numerator and denominator in this q?

$\text{They are simplfying the inequality so that it is easier to solve.}$
$\text{Starting with }\frac{x^2+x-2}{x^2(x-2)}\le0,\text{ multiplying numerator and denominator}$
$\text{by }x-2\text{ like this: }\frac{x^2+x-2}{x^2(x-2)}\cdot\frac{x-2}{x-2}\text{ is the same as multiplying by 1}$
$\text{but has the effect of turning the denominator into a perfect square,$
$\text{hence making it positive so also factorising we have that }$
$\frac{(x+2)(x-1)(x-2)}{x^2(x-2)^2}\le0\text{ and so }(x+2)(x-1)(x-2)\le0.$
$\text{Alternatively you could just multiply }\frac{x^2+x-2}{x^2(x-2)}\le0\text{ by }x^2(x-2)^2\text { to}$
$\text{get the same thing because multiplying by squares preserves the}$
$\text{direction of inequalities.}$

Mechanics Q (1 Viewer)

krypticlemonjuice

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