Question

Solve for x : x
^2 (x-1 )(x-2) ≤ 0 ​

Answer 1

$\bf \huge \boxed{Answer}$

$\bf \: {Let's \: solve \: your \: inequality \: step-by-step.}$

$\bf \: {x}^{2} (x - 1)(x - 2) \: ≤ \: 0$

$\bf \: {x}^{4} - {3x}^{3} + {2x}^{2} ≤ \: 0$

$\bf \: Let's \: find \: the \: critical \: points \: of \: the \: inequality.$

$\bf \: {x}^{4} - {3x}^{3} + {2x}^{2} = 0$

$\bf \: {x}^{2} (x - 1)(x - 2) \: = 0 \bigg(Factor \: left \: side \: of \: equation \bigg)$

$\bf \: {x}^{2} = 0 \: \: \: or \: \: \: x - 1 \: = \: 0 \: \ \: \: or \: \: x - 2 \: = \: 0 \: \bigg(set \: factors \: equal \: to \: 0 \bigg)$

$\bf \: x=0 or x=1 or x=2$

$\bf \: Check$ $\bf \: intervals$ $\bf \: in$ $\bf \: between$ $\bf \: critical$ $\bf \: points$ . $\bigg( \bf \: Test \: values$ $\bf \: in$ $\bf \: the$ $\bf \: intervals$ $\bf \: to$ $\bf \: see$ $\bf \: if$ $\bf \: \bf \: they$ $\bf \: work. \bigg)$

$\bf \: x \: ≤ \: 0 \bigg(Doesn't \: work \: in \: original \: inequality \bigg)$

$\bf \: 0≤x≤1 \bigg(Doesn't \: work \: in \: original \: inequality \bigg)$

$\bf \: 1≤x≤2 \bigg(Works \: in \: original \: inequality \bigg)$

$\bf \: x≥2 \bigg(Doesn't \: work \: in \: original \: inequality \bigg)$

$\bf \underline {Answer:}$

$\bf \: 1≤x≤2 or x=0$

$\bf \underline { Hope \: it's \: helps \: you }$