# What misstep am I making in trying to simplify $\frac{18 - \frac 7 {3x}} {\frac 7 {18x} - 3}$?

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?

• You should post your question using MathJax. Sep 3, 2021 at 4:02
• @Ilovemath Peace. Sure, how do I go about doing that? Sep 3, 2021 at 4:03
• math.meta.stackexchange.com/questions/5020/… Sep 3, 2021 at 4:06

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 :)

It doesn't need to be this complicated. multilply the numerator and denominator by $$18x$$ leaving $$\frac{18 - \frac 7 {3x}} {\frac 7 {18x} - 3} =\frac{324x- 42} {7 - 54x}=-6$$

obviously you are a middle school student so well done for asking for help

• First, thank you. I recently just learned this method that you utilized but even the professor stated that they don't like it so we strayed away from it. Much appreciated beloved. Sep 3, 2021 at 4:14
• @יהודה What method does your prof prefer? How I see it, this is the most intuitive way Sep 3, 2021 at 4:20

$$\require{cancel}$$ As we discussed about technique in another post

$$\dfrac{18 - \dfrac 7 {3x}} {\dfrac 7 {18x} - 3} =\dfrac{\cancel{54x-7}}{3x } \cdot \dfrac{18x}{(-1)(\cancel{54x-7})} =-6$$

• I have no idea when I see it done it's crystal clear but when I am trying to do it alone, its very challenging. I'm like "these exercises aren't like the examples given in the book!!!" lol. Once again thank you. Sep 6, 2021 at 23:21
• @יהודה Do the LCD thing on the numerator and denominator separately. Then you can multiply the numerator by the inverted denominator. Sep 6, 2021 at 23:29