Solve: 2/(3x) = 8/(x + 6) - 2/3
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We want to solve for x in:
`2/(3x) = 8/(x+6) - 2/3`
To do so, we first multiply everything by the least common denominator: In this case that would be `3x (x+6)` . The denominator 3 is already factored in here since `3x` is a common denomintor of the first fraction and the last term.
Multiplying this out:
`2(x+6) = 8(3x) - 2(x)(x+6)`
Then, we have to expand and multiply out everything to get:
`2x + 12 = 24x - 2x^2 - 12x`
`2x^2 -10x + 12 = 0`
Dividing both sides by 2:
`x^2 - 5x + 6 = 0`
This is a quadratic equation that is easily factorable to:
`(x - 2)(x-3) = 0`
Using the zero product property:
`x = 2` or `x = 3` .
Hence, the answers are x = 2 or x = 3.
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TO check, we simply substitute:
If x = 2:
`2/(3*2) = 8/(2+6) - 2/3`
`2/6 = 8/8 - 2/3`
`1/3 = 1 - 2/3`
`1/3 = 1/3` which is true.
If x = 3:
`2/(3*3) = 8/(3+6) - 2/3`
`2/9 = 8/9 - 2/3`
`2/9 = 8/9 - 6/9`
`2/9 = 2/9` which is true!
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