Explain how you calculate the restricted value(s) of x for the equation the fraction the binomial x plus 3 over the binomial x plus 5 plus the fraction x over 6 equals the fraction 5 over 6 and what those value(s) represent. equation included.

Question

Explain how you calculate the restricted value(s) of x for the equation the fraction the binomial x plus 3 over the binomial x plus 5 plus the fraction x over 6 equals the fraction 5 over 6 and what those value(s) represent.  equation included.

anonymous · Answer

$$\frac{ x+3 }{ x+5 }+\frac{ x }{ 6 }=\frac{ 5 }{ 6 }$$

anonymous · Answer

@Hero ?

hero · Answer

The restricted value of x occurs where the denominator cannot equal zero, so to find this restricted value solve x + 5 = 0.

anonymous · Answer

x=-5

hero · Answer

yes, correct.

anonymous · Answer

then it can be anything except -5?

hero · Answer

Yes, correct.

anonymous · Answer

ok thanks!

anonymous · Answer

one more question.  would you use cross-multiplication or LCD to solve it?

anonymous · Answer

you there?

anonymous · Answer

@hero?

hero · Answer

If you know what you're doing, you could use either method.

anonymous · Answer

which one is easier?

hero · Answer

That depends on how familiar you are with either method. But in general, the easier method tends to be the method that requires the least amount of steps to solve.

anonymous · Answer

which is...?

hero · Answer

I'd say it's different for different people. Me personally, I would use the cross multiplication method after putting the equation in proper form.

anonymous · Answer

how would you do that again?

hero · Answer

There's a particular reason for this. If you subtracted 5/6 from both sides you would get:

$$\frac{x + 3}{x + 5} + \frac{x}{6} - \frac{5}{6} = 0$$

Which becomes:
$$\frac{x + 3}{x + 5} + \frac{x - 5}{6} = 0$$

Then after subtracting (x-5)/6 from both sides:
$$\frac{x + 3}{x + 5} =- \frac{x - 5}{6}$$

You would cross multiply to get
$$6(x + 3) = -(x - 5)(x + 5)$$

The right hand side represents a difference of squares:
$$6x + 18 = -(x^2 - 25)$$
$$6x + 18 = -x^2 + 25$$

From there, you would re-arrange it into a quadratic equation then solve.

anonymous · Answer

how would it look once it was finished?

anonymous · Answer

@Hero

hero · Answer

Have you attempted to finish solving it?

anonymous · Answer

yesh but you cant put it into quadratic equations because it doesnt have a  ^2

hero · Answer

It doesn't? Are you sure about that?

anonymous · Answer

oh sorry.  didnt see that!  derp

anonymous · Answer

-x^2-6x+7 right?

hero · Answer

Don't forget to set it equal to zero. You want to make sure the leading term is non-negative. It is easier to factor this way. What you do is this:

$$-x^2 - 6x + 7 = 0$$

Factor out -1:
$$-(x^2 + 6x - 7) = 0$$
Then divide both sides by -1
$$x^2 + 6x - 7 = 0$$

Now you should be able to factor relatively easy

anonymous · Answer

ok thanks! gtg

hero · Answer

Okay, good luck with everything :)