Q4.) Solve the equation by the method of your choice.

Hey there!

First, you must find the common denominator of the equation. 

To find the common denominator, first take a look at all the factors of each denominator:

x: x
x+4: x+4
6: 2*3

Next, because you do not have a common multiple in the denominators, you would multiply the denominators together to create one common multiple:
6(x)(x+4). This would be used to remove the fractions from the equation to make it easier to solve. 

Now, multiply the common multiple 6(x)(x+4) to the entire equation:
(6(x)(x+4)) × ([tex] \frac{1}{x} [/tex] + [tex] \frac{1}{x+4} [/tex] = [tex] \frac{1}{6} [/tex])

When you multiply the factor to the equation, the x in the common factor would cancel out the x in [tex] \frac{1}{x} [/tex] resulting in just 6(x+4).

The x+4 would cancel out the x+4 factor in [tex] \frac{1}{x+4} [/tex] resulting in just 6x.

The 6 would cancel out the 6 in [tex] \frac{1}{6} [/tex] resulting in just x(x+4).

As a result, when you multiply the common factor 6(x)(x+4) to [tex] \frac{1}{x} [/tex] + [tex] \frac{1}{x+4} [/tex] = [tex] \frac{1}{6} [/tex], you will get
(6(x+4))+ 6x = x(x+4)

Now, simplify the equation further:
(6(x+4))+ 6x = x(x+4)
6x+24+6x=x^2+4x   (I have distributed the values in the parentheses)
12x+24=x^2 + 4x   (I combined like terms on both sides)
0=x^2-8x-24   (I have moved all terms to one side so that the x values can                              be solved for using the quadratic formula)

Because this quadratic does not factor evenly, we must use the quadratic formula in order to find the exact x values:
x=[tex] \frac{-b±\sqrt{b^2-4ac} }{2a} [/tex]

Your a value is 1, your b value is -8, and your c value is -24:
x=[tex] \frac{-(-8) ±\sqrt{(-8)^2-4(1)(-24)} }{2(1)} [/tex]
x=[tex] \frac{8 ±\sqrt{64+96} }{2} [/tex]
x=[tex] \frac{8 ±\sqrt{160} }{2} [/tex]
x=[tex] \frac{8 ±4\sqrt{10} }{2} [/tex]
x=4±2[tex] \sqrt{10} [/tex]

Therefore, your x values are 4+2[tex] \sqrt{10} [/tex] and 4-2[tex] \sqrt{10} [/tex]