Question

subtract,
\\(\frac{x}{5x - 8} - \frac{x + 4}{6x}\\)
simplify your answer as much as possible.

Explanation:

Step1: Find the common denominator

The denominators are \(5x - 8\) and \(6x\), so the common denominator is \(6x(5x - 8)\).
Rewrite each fraction with the common denominator:
\(\frac{x}{5x - 8}=\frac{x\cdot6x}{(5x - 8)\cdot6x}=\frac{6x^{2}}{6x(5x - 8)}\)
\(\frac{x + 4}{6x}=\frac{(x + 4)(5x - 8)}{6x(5x - 8)}\)

Step2: Subtract the fractions

\(\frac{6x^{2}}{6x(5x - 8)}-\frac{(x + 4)(5x - 8)}{6x(5x - 8)}=\frac{6x^{2}-(x + 4)(5x - 8)}{6x(5x - 8)}\)

Step3: Expand and simplify the numerator

First, expand \((x + 4)(5x - 8)\):
\((x + 4)(5x - 8)=5x^{2}-8x + 20x - 32=5x^{2}+12x - 32\)
Now, subtract this from \(6x^{2}\):
\(6x^{2}-(5x^{2}+12x - 32)=6x^{2}-5x^{2}-12x + 32=x^{2}-12x + 32\)

Step4: Factor the numerator (if possible)

Factor \(x^{2}-12x + 32\):
We need two numbers that multiply to \(32\) and add to \(-12\). The numbers are \(-8\) and \(-4\).
So, \(x^{2}-12x + 32=(x - 8)(x - 4)\)

Step5: Write the final simplified form

The fraction becomes \(\frac{(x - 8)(x - 4)}{6x(5x - 8)}\) (we can check if further simplification is possible, but in this case, there are no common factors between the numerator and the denominator)

Answer:

\(\frac{x^{2}-12x + 32}{6x(5x - 8)}\) (or \(\frac{(x - 8)(x - 4)}{6x(5x - 8)}\))