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Question

((x-1)/(x+5)) + ((x-3)/(x+2)) = x+13

Answer 1

((x - 1) / (x + 5)) + ((x - 3) / (x + 2)) = x + 13
((x - 1)(x + 2) / (x + 5)) + (x - 3) = (x + 13)(x + 2)
(x - 1)(x + 2) + (x - 3)(x + 5) = (x + 13)(x + 2)(x + 5)
x^2 + x - 2 + x^2 + 2x - 15 = (x + 13)(x^2 + 7x + 10)
2x^2 + 3x - 17 = x^3 + 20x^2 + 101x + 130
x^3 + 18x^2 + 98x + 147 = 0
(x + 7)(x^2 + 11x + 21)
x = -7, (- 11 - √(121 - 84))/2, (- 11 + √(121 - 84))/2
x = -7, (- 11 - √37)/2, (- 11 + √37)/2
x = -7, ( - 11 - 6.082763 ) / 2, ( - 11 + 6.082763 ) / 2
x = - 7, - 8.541381, - 2.458619

Answer 2

Before accepting any of the proposed answers, be sure to double-check them.

For example, 10 is NOT a solution:
|--? ( (10-1) / (10+5) ) + ( (10-3)/(10+2) ) ?=? (10)+13
|--? (9/15) + (7/12) ?=? 23
No: the left side is smaller than 2, the right side is big.

For example, -4 is NOT a solution:
|--? ( (-4-1) / (-4+5) ) + ( (-4-3)/(-4+2) ) ?=? (-4)+13
|--? (-5/1) + (-7/2) ?=? 9
No: The left side is negative, the right side is positive.

For example, 3 is NOT a solution:
|--? ( (3-1) / (3+5) ) + ( (3-3)/(3+2) ) ?=? (3)+13
|--? (2/8) + 0/5 ?=? 16
No: The left side is a fraction, the right side is an integer.


Here's how to solve the equation from scratch.

|--0: ((x-1)/(x+5)) + ((x-3)/(x+2)) = x+13

First, note that the equation is undefined for x=-5 and x=-2.

Next, note that the left side of the equation is a tinch greater than 2 for most values of x. Therefore a value of x several shades larger than -11 should make the right side sorta kinda close to an answer.

Next, note that the left side can become bigger than 2 in only two ways:
a. Either x is rather smaller than -6 (making the first term big and positive),
b. Or else x is close to -2.5 (making the second term big and positive).

So we have three real solutions to find, and the solution set looks something like this:
~ { -8??, -7?, -2.5? } (very roughly)


You can now go one of two ways:
Solve the equation exactly.
Solve the equation numerically.

To solve it exactly, you first clear the denominators by multiplying both sides by (x+2)(x+5):
|--1: ((x-1)(x+2)) + (x-3)(x+5) = (x+13)(x+2)(x+5)

Multiply through:
(x² + x - 2) + (x² + 2x -15) = (x + 13) * (x² + 7x + 10)
2x² + 3x - 17 = x^3 + 20x² +101x + 130
|--2: x^3 + 18x² + 98x + 147 = 0

Nasty-looking, but maybe (x+7) is a factor.
Try it:
|--? ( (-7-1) / (-7+5) + ( (-7-3)/(-7+2) ) ?=? (-7) + 13
|--? (-8/-2) + (-10/-5) ?=? 6
|--? 4 + 2 ?=? 6
Yes: (x+7) is a factor -- yay!

Divide through:
|--3: (x² + 11x +21) (x+7)

Quadratic formula:
x = ( -11 ± √(121-84) ) / 2
x = ½(-11 ± √37)


Is this plausible? Consider √37 ~= 6, and try it:
Solution set: { ~-8.5, -7, -2.5 } (approximately)
That matches well with the proposed approximate set:
{ -8??, -7?, -2.5? } (very roughly)

Double-check the two new answers numerically.

First, x = ½(-11 - √37) ~= -8.541381
|--? ( (-9.54)/(-3.54) ) + ( (-11.54)/(-6.54) ) ?=? (-8.54) + 13
|--? 2.694254 + 1.764365 ?=? 4.458619
Yes.

Next, ½(-11 + √37) ~= -2.458619
|--? ( (-3.46)/(2.54) ) + ( (-5.46)/(-0.46) ) ?=? (-2.46) + 13
|--? -1.360921 + 11.902302 ?=? 10.541381
Yes.


So the solution set is exactly:
|--! { ½(-11-√37), -7, ½(-11+√37) }

This is approximately equal to:
|--~ { -8.541381, -7, -2.458619 }

Answer 3

i got so far and then it didnt seem to work

X - 1 X - 3
------ + -------- = X + 13
X + 5 X + 2

1) Find Least Common Denominator

LCD: (X+5)(X+2)

2) Multiply LCD to the left side. For example, multiply LCD to (X-1)(X-5). Since the LCD has an X+5 in it the denominator cancels. You're left multiplying the numerator (X-1) to (X+2)(1) which is (X-1)(X+2). Do this for the next part.

After you've done this you should have:

(X-1)(X+2) + (X-3)(X+5) = X+13

3) Foil and then Add

X^2 + X - 2 <--first part

X^2 + 2X - 15 <-- second part

Add: X^2 + X - 2 + X^2 + 2X - 15

2X^2 + 3X - 17

Set equal to X+13

2X^2 + 3X - 17 = X + 13
2X^2 + 2X - 30 = 0

Answer 4

((x-1)/(x+5)) + ((x-3)/(x+2)) = x+13
(=) [x^2 + 5X-X - 5] + [ x^2 + 2x-3x -6] = x+13
(=) x^2 + 4x -5 + x^2 - x -6 = x+13
(=) 2(x^2) +3x - 11 = x+13
(=) 2(x^2) +3x-x -11-13 = 0
(=) 2(x^2) +2x -24 = 0
(=) x^2 + x - 12 = 0
(=) (x+4) (x-3) = 0
(=) x+ 4 = 0 or (=) x-3 = 0
(=) x = -4 or (=) x = 3
two answers : 3 and -4

Answer 5

x=10 Combine like terms.

Answer 6

x=10 i think...lol

Answer 7

hmmm, this as far as i can get...2x-squared = -2x +24

Answer 8

x = -3.91427,
x = -2.04287 + 2.79894 i,
x = -2.04287 - 2.79894 i