How i get the $x$ of the equation

Question

I have this:

$\frac{32}{20 - \frac{16}{5-\frac{5}{x}}}$ = 1

My main problem is that I want to multiply by something on both sides.

I will move $(5-\frac{5}{x})$ to side of 1, then:

$\frac{32}{20-16} = 1(5-\frac{5}{x}$)

$8 = 5 -\frac{5}{x}$ , now multiply by $x$

$8x = 5x - 5 $

$3x = 5$

$x = 5/3$

But according to symbolab it's $x = 15/19$, could you help me what I'm wrong about? I have done a lot of exercises, but I have a hard time clearing the $ x $ when it is very low.

Answer 1

This step is uncorrect

$$\frac{32}{20 - \frac{16}{5-\frac{5}{x}}}=1\implies \color{red}{\frac{32}{20-16} = 1(5-\frac{5}{x})}$$

following this way it should be indeed (for $x\neq 1$)

$$\frac{32}{20 - \frac{16}{5-\frac{5}{x}}}=1\implies \frac{1}{(5-\frac{5}{x})}\cdot\frac{32}{20 - \frac{16}{5-\frac{5}{x}}}=\frac{1}{(5-\frac{5}{x})}\cdot 1\implies \frac{32}{20(5-\frac{5}{x})-16} = \frac{1}{(5-\frac{5}{x})}$$

and then

$$\implies 32(5-\frac{5}{x})={20(5-\frac{5}{x})-16}\\\implies160x-160=100x-100 -16x\implies76x=60\implies x=\frac{15}{19}$$

Answer 2

What you've done when you multiply both sides by $(5-\frac{5}{x})$ is wrong. If you multiply what you've got on the left hand side by that then you do not get what you say that you get.

I suggest instead multiplying both sides by the denominator: $20 - \frac{16}{5-\frac{5}{x}}$.

Answer 3

Your first step is incorrect.

You cannot "move" $5-\dfrac 5x$ to one side, you are taking it out incorrectly of your fraction.

I would prefer you multiply a term on the numerator and denominator of your fraction to cancel terms out (this way the fraction also remains the same). For instance, to simplify:

$$\dfrac {16}{5-\frac 5x}=\dfrac {16}{5-\frac 5x}\cdot \dfrac xx=\dfrac{16x}{5x-5}$$

Answer 4

Unfortunately you can't move the $5-{5\over x}$ out of the left side because it is part of a "family" which is the denominator of the bigger fraction.

$$\frac{32}{20 - \frac{16}{5-\frac{5}{x}}}=1$$

If we were to put parenthesis around the terms, it would be clearer why we can't do what you did:

$$\frac{32}{\Big(20 - \frac{16}{\big(5-\frac{5}{x}\big)}\Big)}=1$$

As you can see, the expression $5-{5\over x}$ is "stuck" inside the bigger parenthesis on the bottom, and cannot be taken out.

Having that said, let's solve the equation correctly:

Multiply both sides by the bigger parenthesis denominator:

$$32=20 - \frac{16}{5-\frac{5}{x}}$$ Subtract both sides by 20: $$12=- \frac{16}{5-\frac{5}{x}}$$ $$-12=\frac{16}{5-\frac{5}{x}}$$

Multiply by the "middle" denominator: $$-12\bigg({5-\frac{5}{x}}\bigg)=16$$ $$-60+\frac{60}{x}=16$$ Add 60 to both sides: $$\frac{60}{x}=76$$ Multiply by x: $$76x=60$$ Divide by 76: $$x=\frac{60}{76}$$ $$x=\frac{15}{19}$$ DONE! :D