Question

♠️25 points ♠️

‼️#maths_PrObLeMs‼️

Solve for x: 5/(x+1) + 5/(1-x) = 26​

Answer 1

Answer:

$\frac{5}{\left(x+1\right)}+\frac{5}{\left(1-x\right)}=26\quad:\quad x=\frac{2\sqrt{26}}{13},\:x=-\frac{2\sqrt{26}}{13}$

Step-by-step explanation:

$\frac{5}{\left(x+1\right)}+\frac{5}{\left(1-x\right)}=26$

$\gray{\mathrm{Remove\:the\:parentheses}}$

$\frac{5}{x+1}+\frac{5}{1-x}=26$

$\black{\mathrm{Find\:Least\:Common\:Multiplier\:of\:}x+1,\:1-x:}$

$x+1,\:1-x$

$\gray{\mathrm{Compute\:an\:expression\:comprised\:of\:factors\:that\:appear\:either\:in\:}x+1\mathrm{\:or\:}1-x}$

$=\left(x+1\right)\left(-x+1\right)$

$\gray{\mathrm{Multiply\:by\:LCM=}\left(x+1\right)\left(-x+1\right)}$

$\frac{5}{x+1}\left(x+1\right)\left(-x+1\right)+\frac{5}{1-x}\left(x+1\right)\left(-x+1\right)=26\left(x+1\right)\left(-x+1\right)$

$\gray{\mathrm{Simplify}}$

$5\left(-x+1\right)+5\left(x+1\right)=26\left(x+1\right)\left(-x+1\right)$

$\black{\mathrm{Solve\:}\:5\left(-x+1\right)+5\left(x+1\right)=26\left(x+1\right)\left(-x+1\right):}$

$5\left(-x+1\right)+5\left(x+1\right)=26\left(x+1\right)\left(-x+1\right)$

$\gray{\mathrm{Expand\:}5\left(-x+1\right)+5\left(x+1\right):\quad 10}$

$\gray{\mathrm{Expand\:}26\left(x+1\right)\left(-x+1\right):\quad 26-26x^2}$

$10=26-26x^2$

$\gray{\mathrm{Switch\:sides}}$

$26-26x^2=10$

$\gray{\mathrm{Subtract\:}26\mathrm{\:from\:both\:sides}}$

$26-26x^2-26=10-26$

$\gray{\mathrm{Simplify}}$

$-26x^2=-16$

$\gray{\mathrm{Divide\:both\:sides\:by\:}-26}$

$\frac{-26x^2}{-26}=\frac{-16}{-26}$

$\gray{\mathrm{Simplify}}$

$x^2=\frac{8}{13}$

$\gray{\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}}$

$x=\sqrt{\frac{8}{13}},\:x=-\sqrt{\frac{8}{13}}$

$\black{\sqrt{\frac{8}{13}}}$

$\gray{\mathrm{Apply\:radical\:rule\:}\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0}$

$=\frac{\sqrt{8}}{\sqrt{13}}$

$\gray{\sqrt{8}=2\sqrt{2}}$

$=\frac{2\sqrt{2}}{\sqrt{13}}$

$\gray{\mathrm{Rationalize\:}\frac{2\sqrt{2}}{\sqrt{13}}:\quad \frac{2\sqrt{26}}{13}}$

$=\frac{2\sqrt{26}}{13}$

$\black{-\sqrt{\frac{8}{13}}}$

$\gray{\mathrm{Simplify}\:\sqrt{\frac{8}{13}}:\quad \frac{2\sqrt{2}}{\sqrt{13}}}$

$=-\frac{2\sqrt{2}}{\sqrt{13}}$

$\gray{\mathrm{Rationalize\:}-\frac{2\sqrt{2}}{\sqrt{13}}:\quad -\frac{2\sqrt{26}}{13}}$

$=-\frac{2\sqrt{26}}{13}$

$x=\frac{2\sqrt{26}}{13},\:x=-\frac{2\sqrt{26}}{13}$

Answer 2

$\huge\sf\underline{Answer}$
$<b><i>$
x=0.784465 or x=−0.784465

$\frac{5}{x + 1} + \frac{5}{1 - x} = 26 \\ \frac{5}{x + 1} + \frac{5}{ - x + 1} = 26$

✔️Multiply all the terms by (x+1) (-x+1) and cancel✔️

$5( - x + 1) + 5(x + 1) = 26(x + 1)( - x + 1) \\ 10 = 26 {x}^{2} + 26$

✔️Simplify both sides of the equation✔️
$10 + 26 {x}^{2} = - 26 {x}^{2} + 26 + 26 {x}^{2}$

✔️Add 26^2 both the sides✔️

$26 {x}^{2} + 10 = 26 \\ 26 {x}^{2} + 10 - 10 = 26 - 10$

✔️Subtract 10 both the sides✔️

$26 {x}^{2} = 16 \\ \frac{26 {x}^{2} }{26} = \frac{16}{26}$

✔️Divide both sides 26✔️

${x}^{2} = \frac{8}{13} \\ x = \sqrt{ \frac{8}{13} }$

✔️0.784465,−0.784465✔️

#Regards...❤️