Question

Step-by-step solution to:

`- (3)/(4x - 7) = (x)/(4x - 3)`

Answer 1

The key to solving this problem is recognizing it is a proportion and can be solved using the technique called cross-multiplication. This means that you are multiplying the numerator of each side by the denominator of the other side.


`(-3)/(4x-7) = (x)/(4x-3)`

You multiply the numerator of the first side by the denominator of the second and the denominator of the first by the numerator of the second.


`-3(4x-3)=x(4x-7)`

Simplify.


`-12x+9=4x^2-7x`


`4x^2+5x-9=0`

Factor.

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

Set each factor equal to zero and solve for x.


`4x+9=0 \\ x=-9/4`


`x-1=0 \\ x=1`

Finally, you want to look at the original expression and make sure that these solutions don't violate any restrictions on division by zero.

For the right side, we see that
`4x-7` ≠
`0`, so
`x` ≠
`7/4`. For the left side, we see that
`4x-3` ≠
`0`, so
`x` ≠
`3/4`. This means our solutions are valid.

x = -9/4, x = 1