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Is-absolute-value-of-x-linear-equation-

Question Number 215919 by abdelsalam last updated on 21/Jan/25 $$\:\:\:{Is}\:{absolute}\:{value}\:{of}\:{x}\:{linear}\:{equation}\:?\: \\ $$$$ \\ $$ Commented by A5T last updated on 21/Jan/25 $$\mathrm{General}\:\mathrm{linear}\:\mathrm{equation}:\:\mathrm{y}=\mathrm{mx}+\mathrm{c} \\ $$$$\mathrm{where}\:\mathrm{m}\:\mathrm{is}\:\mathrm{the}\:\mathrm{constant}\:\mathrm{slope} \\…

for-any-natural-numbers-m-n-then-m-n-or-m-lt-n-or-m-gt-n-prove-

Question Number 215939 by mokys last updated on 21/Jan/25 $${for}\:{any}\:{natural}\:{numbers}\:{m},{n}\:{then}\:{m}={n}\:{or}\:{m}<{n}\:{or}\:{m}>{n}\:?\:{prove} \\ $$ Answered by MrGaster last updated on 22/Jan/25 $$\forall{m},{n}\in\mathbb{N} \\ $$$${m}={n}\vee{m}<{n}\vee{m}>{n} \\ $$$$\because\mathbb{N}\subset\mathbb{R}\wedge\mathbb{R}\:\mathrm{is}\:\mathrm{totally}\:\mathrm{ordered} \\…

Let-be-a-hyperbola-with-foci-F-1-and-F-2-eccentricity-e-M-is-an-arbitrary-point-on-Let-x-MF-1-F-2-y-MF-2-F-1-Prove-that-cos-x-cos-y-1-cos-x-cos-y-2e-e-2-1-

Question Number 215828 by CrispyXYZ last updated on 19/Jan/25 $$\mathrm{Let}\:\Gamma\:\mathrm{be}\:\mathrm{a}\:\mathrm{hyperbola}\:\mathrm{with}\:\mathrm{foci}\:{F}_{\mathrm{1}} \:\mathrm{and}\:{F}_{\mathrm{2}} ,\: \\ $$$$\mathrm{eccentricity}\:{e}.\:{M}\:\mathrm{is}\:\mathrm{an}\:\mathrm{arbitrary}\:\mathrm{point}\:\mathrm{on}\:\Gamma. \\ $$$$\mathrm{Let}\:{x}=\angle{MF}_{\mathrm{1}} {F}_{\mathrm{2}} ,\:{y}=\angle{MF}_{\mathrm{2}} {F}_{\mathrm{1}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\frac{\mid\:\mathrm{cos}\:{x}\:−\:\mathrm{cos}\:{y}\:\mid}{\mathrm{1}\:−\:\mathrm{cos}\:{x}\:\mathrm{cos}\:{y}}\:=\:\frac{\mathrm{2}{e}}{{e}^{\mathrm{2}} +\mathrm{1}}. \\ $$ Answered…

Let-g-x-3f-x-3-f-3-x-and-f-x-gt-0-for-all-x-0-3-If-g-is-decreasing-in-0-and-increasing-in-3-then-find-8-

Question Number 215760 by MATHEMATICSAM last updated on 17/Jan/25 $$\mathrm{Let}\:{g}\left({x}\right)\:=\:\mathrm{3}{f}\left(\frac{{x}}{\mathrm{3}}\right)\:+\:{f}\left(\mathrm{3}\:−\:{x}\right)\:\mathrm{and}\: \\ $$$${f}\:''\left({x}\right)\:>\:\mathrm{0}\:\mathrm{for}\:\mathrm{all}\:{x}\:\in\:\left(\mathrm{0},\:\mathrm{3}\right).\:\mathrm{If}\:{g}\:\mathrm{is}\: \\ $$$$\mathrm{decreasing}\:\mathrm{in}\:\left(\mathrm{0},\:\alpha\right)\:\mathrm{and}\:\mathrm{increasing}\:\mathrm{in} \\ $$$$\left(\alpha,\:\mathrm{3}\right)\:\mathrm{then}\:\mathrm{find}\:\mathrm{8}\alpha. \\ $$ Answered by universe last updated on 18/Jan/25…

Tow-men-complet-a-work-in-3hr-three-women-complet-it-in-5hr-and-five-children-complet-it-in-10hr-if-we-consider-among-these-one-man-one-woman-and-one-child-in-how-much-time-they-will-complet-it-t

Question Number 215755 by Davidtim last updated on 17/Jan/25 $${Tow}\:{men}\:{complet}\:{a}\:{work}\:{in}\:\mathrm{3}{hr},\:{three} \\ $$$${women}\:{complet}\:{it}\:{in}\:\mathrm{5}{hr},\:{and}\:{five}\: \\ $$$${children}\:{complet}\:{it}\:{in}\:\mathrm{10}{hr}.\:{if}\:{we}\:{consider} \\ $$$${among}\:{these}\:{one}\:{man},\:{one}\:{woman}\:{and}\:{one} \\ $$$${child}\:{in}\:{how}\:{much}\:{time}\:{they}\:{will} \\ $$$${complet}\:{it}\:{together}? \\ $$ Answered by A5T…