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Solve-2x-x-1-3-




Question Number 178595 by Acem last updated on 18/Oct/22
Solve ((2x)/(x+1))≥ 3
$${Solve}\:\frac{\mathrm{2}{x}}{{x}+\mathrm{1}}\geqslant\:\mathrm{3} \\ $$
Answered by Acem last updated on 19/Oct/22
The safest and surest way to avoid making   mental mistakes:     ((2x)/(x+1))−3≥ 0 ⇔ ((−x−3)/(x+1))≥ 0     ⇔ ((x^(Always try it as positive) +3)/(x+1)) ≤ 0       x     −∞         −3           −1             +∞   x+3              −     0      +             +   x+1              −             −    0       +   Frac.            +     0      −    ∥       +     ⇒ x∈ [−3, −1[
$${The}\:{safest}\:{and}\:{surest}\:{way}\:{to}\:{avoid}\:{making} \\ $$$$\:{mental}\:{mistakes}: \\ $$$$ \\ $$$$\:\frac{\mathrm{2}{x}}{{x}+\mathrm{1}}−\mathrm{3}\geqslant\:\mathrm{0}\:\Leftrightarrow\:\frac{−{x}−\mathrm{3}}{{x}+\mathrm{1}}\geqslant\:\mathrm{0} \\ $$$$ \\ $$$$\:\Leftrightarrow\:\frac{\boldsymbol{{x}}^{{Always}\:{try}\:{it}\:{as}\:{positive}} +\mathrm{3}}{\boldsymbol{{x}}+\mathrm{1}}\:\leqslant\:\mathrm{0} \\ $$$$ \\ $$$$\:\:\:{x}\:\:\:\:\:−\infty\:\:\:\:\:\:\:\:\:−\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\cancel{−\mathrm{1}}\:\:\:\:\:\:\:\:\:\:\:\:\:+\infty \\ $$$$\:{x}+\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:+\:\:\:\:\:\:\:\:\:\:\:\:\:+ \\ $$$$\:{x}+\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\:\:\:\:\:\:\:\:\:\:\:\:\:−\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:+ \\ $$$$\:{Frac}.\:\:\:\:\:\:\:\:\:\:\:\:\cancel{+}\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:−\:\:\:\:\parallel\:\:\:\:\:\:\:\cancel{+} \\ $$$$ \\ $$$$\:\Rightarrow\:\boldsymbol{{x}}\in\:\left[−\mathrm{3},\:−\mathrm{1}\left[\right.\right. \\ $$$$ \\ $$
Commented by Acem last updated on 18/Oct/22
 (a/(b )) (a/(⇛ make it as your opponent that you must not forget))
$$\:\frac{{a}}{\boldsymbol{{b}}\:}\:\frac{{a}}{\Rrightarrow\:\boldsymbol{{make}}\:\boldsymbol{{it}}\:\boldsymbol{{as}}\:\boldsymbol{{your}}\:\boldsymbol{{opponent}}\:\boldsymbol{{that}}\:\boldsymbol{{you}}\:\boldsymbol{{must}}\:\boldsymbol{{not}}\:\boldsymbol{{forget}}} \\ $$

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