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Question Number 23814 by tawa tawa last updated on 06/Nov/17

Commented by tawa tawa last updated on 06/Nov/17

please help.

$$\mathrm{please}\:\mathrm{help}. \\ $$

Answered by mrW1 last updated on 06/Nov/17

⇒x≥−(a/b)  ⇒x≥−(b/c)  ⇒x≥−(c/a)  ⇒x^3 ≥−1  ⇒x≥−1    ⇒x≤(b/a)  ⇒x≤(c/b)  ⇒x≤(a/c)  ⇒x^3 ≤1  ⇒x≤1    x=0 is the solution.

$$\Rightarrow{x}\geqslant−\frac{{a}}{{b}} \\ $$$$\Rightarrow{x}\geqslant−\frac{{b}}{{c}} \\ $$$$\Rightarrow{x}\geqslant−\frac{{c}}{{a}} \\ $$$$\Rightarrow{x}^{\mathrm{3}} \geqslant−\mathrm{1} \\ $$$$\Rightarrow{x}\geqslant−\mathrm{1} \\ $$$$ \\ $$$$\Rightarrow{x}\leqslant\frac{{b}}{{a}} \\ $$$$\Rightarrow{x}\leqslant\frac{{c}}{{b}} \\ $$$$\Rightarrow{x}\leqslant\frac{{a}}{{c}} \\ $$$$\Rightarrow{x}^{\mathrm{3}} \leqslant\mathrm{1} \\ $$$$\Rightarrow{x}\leqslant\mathrm{1} \\ $$$$ \\ $$$${x}=\mathrm{0}\:{is}\:{the}\:{solution}. \\ $$

Commented by tawa tawa last updated on 06/Nov/17

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

Commented by Joel577 last updated on 07/Nov/17

If x ≥ −1 and x ≤ 1, why x = 0 is only the solution?

$$\mathrm{If}\:{x}\:\geqslant\:−\mathrm{1}\:\mathrm{and}\:{x}\:\leqslant\:\mathrm{1},\:\mathrm{why}\:{x}\:=\:\mathrm{0}\:\mathrm{is}\:\mathrm{only}\:\mathrm{the}\:\mathrm{solution}? \\ $$

Commented by mrW1 last updated on 07/Nov/17

for the solution we don′t need to know  that −1≤x≤1.  we know that x=0 is a solution.  we know theLHS is strictly increasing  and the RHS is strictly decreasing.  that means if there is a solution then  this solution is the only solution.

$${for}\:{the}\:{solution}\:{we}\:{don}'{t}\:{need}\:{to}\:{know} \\ $$$${that}\:−\mathrm{1}\leqslant{x}\leqslant\mathrm{1}. \\ $$$${we}\:{know}\:{that}\:{x}=\mathrm{0}\:{is}\:{a}\:{solution}. \\ $$$${we}\:{know}\:{theLHS}\:{is}\:{strictly}\:{increasing} \\ $$$${and}\:{the}\:{RHS}\:{is}\:{strictly}\:{decreasing}. \\ $$$${that}\:{means}\:{if}\:{there}\:{is}\:{a}\:{solution}\:{then} \\ $$$${this}\:{solution}\:{is}\:{the}\:{only}\:{solution}. \\ $$

Commented by Joel577 last updated on 07/Nov/17

understood, Sir. Thank you

$${understood},\:{Sir}.\:{Thank}\:{you} \\ $$

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