Simplify $\dfrac{18 – \dfrac 7 {3x}} {\dfrac 7 {18x} – 3}$?
I'm having a hard time simplifying this particular expression and am seeking any type of assistance in solving it.
In the expression
$$\frac{18 – \dfrac 7 {3x}} {\dfrac 7 {18x} – 3}$$ I split the problems into two separate entities.
For the numerator, I get $3x$ for the LCD and then rewrite the fraction as $$54x-7\frac 7 {3x}$$ As for the denominator, I get $18x$ for the LCD and then rewrite the fraction as
$$7-\frac{54x}{18x}$$
When I begin to divide, I switch the sign from division to multiplication and swap the numerator with the denominator ($7-\frac{54x}{18x}$ becomes $\frac{18x}{7-54x}$).
The product I get is $$\frac{972x^2-126x}{21x – 162x^2}$$
When I simplify I get $6-6$ which is zero. Is this answer correct?
Best Answer
Using your method, the numerator $N$ is $$N=18-\dfrac{7}{3x}=\dfrac{54x}{3x}-\dfrac{7}{3x}=\dfrac{54x-7}{3x}$$ The denominator is $$D=\dfrac{7}{18x}-3=\dfrac{7}{18x}-\dfrac{54x}{18x}=\dfrac{7-54x}{18x}$$ Thus, the given fraction is $$F=N÷D=\dfrac{54x-7}{3x}÷\dfrac{7-54x}{18x}$$ $$\implies F=\dfrac{54x-7}{3x}×\dfrac{18x}{7-54x}=\boxed{-6}$$
Hope this helps. Ask anything if not clear :)