Question #a71e9

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Question #a71e9

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Answer 1 · 2016-09-19T11:12:13

$x = - \frac{10}{3}$ $x=color(green)(-10/3)$

Explanation:

Given
$XXX \frac{4}{x + 2} + \frac{1}{x + 3} = \frac{8}{x + 2}$ $color(white)("XXX")4/(x+2)+1/(x+3)=8/(x+2)$

Multiplying both sides by $(x + 2) (x + 3)$ $(x+2)(x+3)$
$XXX 4 (x + 3) + 1 (x + 2) = 8 (x + 3)$ $color(white)("XXX")4(x+3)+1(x+2)=8(x+3)$

Simplifying
$XXX 5 x + 14 = 8 x + 24$ $color(white)("XXX")5x+14=8x+24$

$XXX - 3 x = 10$ $color(white)("XXX")-3x=10$

$XXX x = - \frac{10}{3} (= 3 \frac{1}{3})$ $color(white)("XXX")x=-10/3 (=3 1/3)$

Answer 2 · 2016-09-19T11:15:41

$x = - \frac{10}{3}$ $x=-10/3$

Explanation:

Basically we need to isolate $x$ $x$ so I will first multiply both sides by $x + 2$ $x+2$

$4 + \frac{x + 2}{x + 3} = 8$ $4+(x+2)/(x+3)=8$
now move $4$ $4$ over
$\frac{x + 2}{x + 3} = 4$ $(x+2)/(x+3)=4$
now multiply both sides by $x + 3$ $x+3$
$x + 2 = 4 x + 12$ $x+2=4x+12$
Move like terms to each side
$- 3 x = 10$ $-3x=10$
Solve for $x$ $x$
$x = - \frac{10}{3}$ $x=-10/3$

As a final step always plug in and check your answer.

Answer 3 · 2016-09-19T11:22:37

$x = - \frac{10}{3}$ $x=-10/3$
This solution is not the most efficient on purpose: The objective is to demonstrates a basic principle that may be used else ware.

Explanation:

Given: $\frac{4}{x + 2} + \frac{1}{x + 3} = \frac{8}{x + 2}$ $" "4/(x+2)+1/(x+3)=8/(x+2)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$[\frac{4}{x + 2} \times 1] + [\frac{1}{x + 3} \times 1] = [\frac{8}{x + 2} \times 1]$ $[4/(x+2)color(red)(xx1)]+ [1/(x+3)color(blue)(xx1)]" "=" "[8/(x+2)color(green)(xx1)]$

$.$ $color(white)(.)$

$[\frac{4}{x + 2} \times \frac{x + 3}{x + 3}] + [\frac{1}{x + 3} \times \frac{x + 2}{x + 2}] = [\frac{8}{x + 2} \times \frac{x + 3}{x + 3}]$ $[4/(x+2)color(red)(xx(x+3)/(x+3))]+ [1/(x+3)color(blue)(xx(x+2)/(x+2))]=[8/(x+2)color(green)(xx(x+3)/(x+3))]$

$.$ $color(white)(.)$

$[\frac{4 (x + 3)}{(x + 2) (x + 3)}] + [\frac{x + 2}{(x + 2) (x + 3)}] = [\frac{8 (x + 3)}{(x + 2) (x + 3)}]$ $[(4(x+3))/((x+2)(x+3))] +[(x+2)/((x+2)(x+3))] =[(8(x+3))/((x+2)(x+3))]$
$.$ $color(white)(.)$

Multiply both sides by $(x + 2) (x + 3)$ $(x+2)(x+3)$ giving:

$4 (x + 3) + x + 2 = 8 (x + 3)$ $4(x+3)" "+" "x+2" "=" "8(x+3)$

$. 4 x + 12 + x + 2 = 8 x + 24$ $color(white)(.)4x+12" "+" "x+2" "=" "8x+24$

$5 x + 14 = 8 x + 24$ $" "5x+14" "=" "8x+24$

$- 10 = 3 x$ $" "-10" "=" "3x$

$x = - \frac{10}{3}$ $" "x=-10/3$

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