How do you solve `6/x + 7/(x + 3) = 4`?

The first step to answering this is multiplying the whole equation by `x`:

`6 + (7x) / (x+3) = 4x`

Then multiply the whole equation by `x+3` :

`6(x+3) + 7x = 4x(x+3)`

Expanding;

`6x + 18 + 7x = 4x^2 + 12x`

Rearanging:

`4x^2 - x - 18 = 0`

Now we have a standard quadratic equation, applying the quadratic formula or factorising to yeild:

`x= 9/4 , x = -2`

First, multiply each side of the equation by `color(red)(x)(color(blue)(x + 3))` to eliminate the fractions while keeping the equation balanced:

`color(red)(x)(color(blue)(x + 3))(6/x + 7/(x + 3)) = color(red)(x)(color(blue)(x + 3))4`

`(color(red)(x)(color(blue)(x + 3)) xx 6/x) + (color(red)(x)(color(blue)(x + 3)) xx 7/(x + 3)) = (x^2 + 3x)4`

`(cancel(color(red)(x))(color(blue)(x + 3)) xx 6/color(red)(cancel(color(black)(x)))) + (color(red)(x)cancel((color(blue)(x + 3))) xx 7/color(blue)(cancel(color(black)(x + 3)))) = 4x^2 + 12x`

`6(color(blue)(x + 3)) + 7color(red)(x) = 4x^2 + 12x`

`6x + 18 + 7x = 4x^2 + 12x`

`6x + 7x + 18 = 4x^2 + 12x`

`13x + 18 = 4x^2 + 12x`

We can now put the equation in standard form:

`13x - color(red)(13x) + 18 - color(blue)(18) = 4x^2 + 12x - color(red)(13x) - color(blue)(18)`

`0 + 0 = 4x^2 + (12 - color(red)(13))x - 18`

`0 = 4x^2 + (-1)x - 18`

`0 = 4x^2 - 1x - 18`

`4x^2 - 1x - 18 = 0`

We can now factor the left side of the equation as:

`(4x - 9)(x + 2) = 0`

Now, solve each term on the left for `0` to find the solutions to the problem:

Solution 1:

`4x - 9 = 0`

`4x - 9 + color(red)(9) = 0 + color(red)(9)`

`4x - 0 = 9`

`4x = 9`

`(4x)/color(red)(4) = 9/color(red)(4)`

`(color(red)(cancel(color(black)(4)))x)/cancel(color(red)(4)) = 9/4`

`x = 9/4`

Solution 2:

`x + 2 = 0`

`x + 2 - color(red)(2) = 0 - color(red)(2)`

`x + 0 = -2`

`x = -2`

The Solution Are: `x = {-2, 9/4}`

`6/x + 7/(x+3) = 4`

The first thing you want to do is to make the denominators the same. How to do that? Well, let's multiply by one, which shouldn't do anything wrong, right?:

`6/x * 1 + 7/(x+3) * 1 = 4`

Now, any number divided by itself equals one, doesn't it? Use this to your advantage! Let's change the one on the left into `(x + 3)/(x + 3)`:

`6/x * (x + 3)/(x + 3) + 7/(x+3) * 1 = 4`

And the one on the right into `x/x`:

`6/x * (x + 3)/(x + 3) + 7/(x+3) * x/x = 4`

Multiply:

`(6(x + 3))/(x(x + 3)) + (7x)/(x(x + 3)) = 4`

Now we can add!

`(6(x + 3) + (7x))/(x(x + 3)) = 4`

Distribute and simplify:

`(6x + 18 + 7x)/(x^2 + 3x) = 4`

`(13x + 18)/(x^2 + 3x) = 4`

Let's multiply that denominator by itself, to both sides:

`(13x + 18)/(x^2 + 3x) * (x^2 + 3x) = 4 * (x^2 + 3x)`

`13x + 18 = 4x^2 + 12x`

Ahh, this is going to be a quadratic equation! Subtract both sides by `4x^2 + 12x`:

`13x + 18 - (4x^2 + 12x) = 4x^2 + 12x - (4x^2 + 12x)`

`-4x^2 + x + 18 = 0`

Let's complete the square, first dividing everything by `-4`:

`(-4x^2)/(-4) + x/(-4) + 18/(-4) = 0/(-4)`

`x^2 -1/4 (x) - 9/2 = 0`

Now, imagine a square `x^2` and a rectangle glued to its right with length `-1/4` and width `x`. Split that rectangle into half so that the other half glues to the bottom.

`x^2 + 2(-1/8)(x) - 9/2 = 0`

There's a spot on the bottom-right looking like a square can fit. The square would have a side length of `-1/8`, so its area is `(-1/8)^2`. Let's add it (to both sides, of course)!

`x^2 + 2(-1/8)(x) + (-1/8)^2 - 9/2 = 1/64`

Now look at the entire picture. All four shapes form a bigger square with side length `x - 1/8`, so the total area is `(x - 1/8)^2`. Replace all of the shapes with this (another way to look at this is using the rule `a^2 + 2ab + b^2 = (a + b)^2`).

`(x - 1/8)^2 - 9/2 = 1/64`

We're done completing the square; there's only one `x` now, so let's solve! Add both sides by `9/2`:

`(x - 1/8)^2 - 9/2 + 9/2 = 1/64 + 9/2`

`(x - 1/8)^2 = 1/64 + 288/64`

`(x - 1/8)^2 = 289/64`

Take the square root (but be careful! `3^2` and `(-3)^2` both equal `9`, so `sqrt(9) = ±3`):

`sqrt((x - 1/8)^2) = ±sqrt(289/64)`

Power of two and square root cancels out, and we can split the square root into the numerator and denominator:

`x - 1/8 = ±sqrt(289)/sqrt(64)`

`x - 1/8 = ±17/8`

Add `1/8` to both sides:

`x - 1/8 + 1/8 = 1/8 ± 17/8`

`x = 1/8 ± 17/8`

There are really two answers here: a positive and a negative one. We need both:

`x_1 = 1/8 + 17/8 = 18/8 = 9/4 = 2.25`

`x_2 = 1/8 - 17/8 = -16/8 = -2`

There we go! So the solutions are `x = {2.25, -2}`.

`6/x+7/(x+3)=4`

multiply both sides by `x`

`:.6(color(magenta)x/x)+(7(color(magenta)x))/(x+3)=4(color(magenta)x)`

`:.6+(7x)/(x+3)=4x`

multiply both sides by `color(magenta)(x+3`

`:.6color(magenta)((x+3))+7xcancel(color(magenta)((x+3)^1))/cancelcolor(magenta)color(magenta)(x+3)^1=4xcolor(magenta)((x+3)`

`:.6x+18+7x=4x^2+12x`

`:.4x^2+12x=6x+18+7x`

`:.4x^2+12x-6x-7x=18`

`:.4x^2-x-18=0`

`:.(4x-9)(x+2)=0`

`:.4x=9 or x=-2`

`:.color(magenta)(x=9/4 or x=-2`

~~~~~~~~~~~~~~~~

check:-

substitute`color(magenta)(x=-2`

`:.6/(color(magenta)((-2)))+7/((color(magenta)(-2)+3)=4`

`:.-3+7/1=4`

`:.-3+7=4`

`:.color(magenta)(4=4`

substitute`color(magenta)(x=9/4`

`:.6/(color(magenta)(9/4))+7/(color(magenta)(9/4)+3)=4`

`:.(cancel6^color(magenta)2/1xx4/cancel9^color(magenta)3)+7/((2 1/4)+3)=4`

`:.8/3+7/(5 1/4)=4`

`:.8/3+(cancel7^color(magenta)1/1xx4/cancel21^color(magenta)3)=4`

`8/3+4/3=4`

`:.12/3=4`

`:.color(magenta)(4=4`