\[\frac{{x}^{3}}{2}\ge \frac{2}{3}\]
1
Multiply both sides by 22.
x3≥23×2{x}^{3}\ge \frac{2}{3}\times 2

2
Use this rule: ab×c=acb\frac{a}{b} \times c=\frac{ac}{b}.
x3≥2×23{x}^{3}\ge \frac{2\times 2}{3}

3
Simplify  2×22\times 2  to  44.
x3≥43{x}^{3}\ge \frac{4}{3}

4
Take the cube root of both sides.
x≥433x\ge \sqrt[3]{\frac{4}{3}}

5
Use Division Distributive Property: (xy)a=xaya{(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}.
x≥4333x\ge \frac{\sqrt[3]{4}}{\sqrt[3]{3}}

6
Rewrite 44 as 22{2}^{2}.
x≥22333x\ge \frac{\sqrt[3]{{2}^{2}}}{\sqrt[3]{3}}

7
Use this rule: (xa)b=xab{({x}^{a})}^{b}={x}^{ab}.
x≥22333x\ge \frac{{2}^{\frac{2}{3}}}{\sqrt[3]{3}}

Done

Decimal Form: 1.100642
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