How do you do this problem ?

Question

Solve the equation.

(1) / ( x + 7 ) + (5) / ( x + 6) = (-1) / ( x^2 + 13x + 42 )

Answer 1

To add fractions, you must have common denominators. For instance, to add 1/2 + 1/3, you change both into 6ths. This can often times be accomplished by multiplying one of the denominators by the other. Keep in mind that what you do to the bottom of a fraction, you must do to the top as well.

Thus, 1/2 + 1/3 becomes 1(3)/2(3) + 1(2)/3(2) = 3/6 + 2/6 = 5/6.

To answer your algebra question directly, we will apply our arithmetic rule sampled above. Multiply each denominator by the other's, and do the same to the tops.

1(x+6)/(x+7)(x+6) + 5(x+7)/(x+6)(x+7)

Distribute along every top and bottom. This becomes x+6/(x^2 + 13x + 42) + (5x + 35)/(x^2 + 13x + 42). Adding the tops, you end with (6x + 41)/(x^2 + 13x + 42).

So, total equation is: (6x + 41)/(x^2 + 13x + 42) = (-1) / ( x^2 + 13x + 42 )

Since denominators are the same, you can ignore them.

Thus, 6x + 41 = -1. Solving for x, subtract 41 on each side of equal sign, then divide by 6.

6x + 41 = -1
6x = -42
x = -7

To check your answer, plug -7 in for the original x's in your problem. You will discover that the first fraction is impossible, so even though you theoretically have an answer of x = -7, it cannot be that number.

Final answer: Empty set ... or none if you prefer

Answer 2

The least common denominator is (x+7)(x+6), since that's what ( x^2 + 13x + 42 ) factors into:

(1) / ( x + 7 ) + (5) / ( x + 6) = (-1) / ( x^2 + 13x + 42 )

(x+6) / (x+7)(x+6) + 5(x+7) / (x+7)(x+6) = -1 / (x+7)(x+6)

Now that everything has the same denominator, you can combine the numerators, moving -1 over to the left-hand side as +1:

(x + 6 + 5x + 35 + 1) / (x+7)(x+6) = 0
(6x + 42) / (x+7)(x+6) = 0

Normally, you'd ignore the denominator (or, really, multiply both sides by the denominator, which still yields zero on the right-hand side) and get:
6x + 42 = 0
6x = -42
x = -7

However x=-7 is ruled out by the initial equation having (x+7) in the denominator. Try plugging x=-7 into the original equation and see what you get:

(1) / ( x + 7 ) + (5) / ( x + 6) = (-1) / ( x^2 + 13x + 42 )
1 / (-7 + 7) + 5 / (-7 + 6) = (-1) / ((-7)^2 + 13*-7 + 42)
1 / 0 + 5 / -1 = -1 / (49 - 91 + 42)

1 / 0 + -5 = -1 / 0 ?!?

There are no solutions.

Answer 3

multiply both sides by(x+6)(x+7)
this results in

(x+6)+5(x+7)=-1
6x+41=-1
6x=-42
x=-7

Answer 4

(x+6)/(x+7)(x+6)+5(x+7)/(x+6)(x+7)=-1/(x^2+13x+42)
[(x+6+5x+35)/(x^2+13x+42)]=-1/(x^2+13x+42)
6x+41=-1
6x=-42
x=-42/6=-7