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least common denominator (lcd)
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question 16 of 40
subtract and simplify if possible.
\\(\frac{8 - x}{9x + 1} - \frac{3x - 8}{9x + 1}\\)

\\(\frac{8 - x}{9x + 1} - \frac{3x - 8}{9x + 1} = \underline{\quad\quad}\\)

Explanation:

Step1: Subtract numerators (same denominator)

Since the denominators are the same ($9x + 1$), we subtract the numerators: $(8 - x) - (3x - 8)$.

Step2: Simplify the numerator

Simplify $(8 - x) - (3x - 8) = 8 - x - 3x + 8 = 16 - 4x$.

Step3: Factor the numerator

Factor $16 - 4x$ as $-4(x - 4)$ or factor out $4$: $4(4 - x)$. Wait, alternatively, let's re - examine the subtraction:
Wait, $(8 - x)-(3x - 8)=8 - x - 3x + 8=16 - 4x = 4(4 - x)$ or we can also factor out $-4$: $-4(x - 4)$. But let's also check the denominator $9x + 1$.
Wait, another way: $(8 - x)-(3x - 8)=8 - x - 3x + 8 = 16-4x=4(4 - x)$. But we can also write the numerator as $-4x + 16=-4(x - 4)$.
Now, the fraction is $\frac{16 - 4x}{9x + 1}=\frac{-4x + 16}{9x + 1}=\frac{-4(x - 4)}{9x + 1}$ or we can factor out $4$ from the numerator: $\frac{4(4 - x)}{9x + 1}$. But we can also simplify the numerator and denominator (if possible). Wait, but let's do the subtraction again:
Wait, original expression: $\frac{8 - x-(3x - 8)}{9x + 1}=\frac{8 - x - 3x + 8}{9x + 1}=\frac{16-4x}{9x + 1}=\frac{4(4 - x)}{9x + 1}$ or $\frac{-4(x - 4)}{9x + 1}$. But we can also factor the numerator as $-4(x - 4)$ and see if we can simplify. But actually, we can factor out $4$ from the numerator: $\frac{4(4 - x)}{9x + 1}$ or we can write the numerator as $-4x + 16$ and then factor out $-4$: $\frac{-4(x - 4)}{9x + 1}$.
Wait, but let's check the subtraction once more. $(8 - x)-(3x - 8)=8 - x - 3x + 8 = 16-4x = 4(4 - x)$. Now, the fraction is $\frac{4(4 - x)}{9x + 1}$. But we can also factor the numerator as $-4(x - 4)$ and the denominator is $9x + 1$.
Alternatively, we can factor the numerator: $16-4x = 4(4 - x)$ and the denominator is $9x + 1$. But we can also simplify the fraction by factoring the numerator and seeing if there are common factors with the denominator. Since the denominator is $9x + 1$ and the numerator is $4(4 - x)$ (or $-4(x - 4)$), there are no common factors between the numerator and the denominator. Wait, but let's re - do the subtraction:
Wait, $(8 - x)-(3x - 8)=8 - x - 3x + 8 = 16-4x=4(4 - x)$. So the fraction is $\frac{4(4 - x)}{9x + 1}$ or we can factor out $-4$ from the numerator: $\frac{-4(x - 4)}{9x + 1}$.
But we can also simplify the numerator and denominator by dividing numerator and denominator by a common factor (if any). But since $9x + 1$ and $4(4 - x)$ have no common factors, the simplified form is $\frac{16 - 4x}{9x + 1}=\frac{4(4 - x)}{9x + 1}$ or $\frac{-4x + 16}{9x + 1}$. We can also factor the numerator as $-4(x - 4)$ and write the fraction as $\frac{-4(x - 4)}{9x + 1}$.
Wait, another approach:
$\frac{8 - x}{9x + 1}-\frac{3x - 8}{9x + 1}=\frac{(8 - x)-(3x - 8)}{9x + 1}=\frac{8 - x - 3x + 8}{9x + 1}=\frac{16-4x}{9x + 1}=\frac{4(4 - x)}{9x + 1}=\frac{-4(x - 4)}{9x + 1}$
We can also factor the numerator as $-4(x - 4)$ and the denominator is $9x + 1$. There are no common factors between the numerator and the denominator, so the simplified form is $\frac{16 - 4x}{9x + 1}=\frac{4(4 - x)}{9x + 1}$ or we can factor out $4$ from the numerator: $\frac{4(4 - x)}{9x + 1}$ or factor out $-4$: $\frac{-4(x - 4)}{9x + 1}$.
But we can also write the numerator as $-4x + 16$ and then divide numerator and denominator by... Wait, no common factors. So the simplified form is $\frac{16 - 4x}{9x + 1}=\frac{4(4 - x)}{9x + 1}$ or $\frac{-4(x - 4)}{9x + 1}$. We can also factor the numerator as $-4(x - 4)$ and the denominator is $9x + 1$.
Alternatively, we can factor the numerator $16 - 4x$ as $4(4 - x)$ and the denominator is $9x + 1$. So the simplified fraction is $\frac{4(4 - x)}{9x + 1}$ or $\frac{-4(x - 4)}{9x + 1}$. We can also write the numerator as $-4x + 16$ and the fraction is $\frac{-4x + 16}{9x + 1}=\frac{-4(x - 4)}{9x + 1}$.
But let's check the arithmetic again:
$(8 - x)-(3x - 8)=8 - x - 3x + 8 = 16-4x$. So the fraction is $\frac{16 - 4x}{9x + 1}$. We can factor out $4$ from the numerator: $\frac{4(4 - x)}{9x + 1}$ or factor out $-4$: $\frac{-4(x - 4)}{9x + 1}$.
We can also simplify the numerator and denominator by dividing numerator and denominator by a common factor. Since $9x + 1$ and $4(4 - x)$ have no common factors, the simplified form is $\frac{4(4 - x)}{9x + 1}$ or $\frac{-4(x - 4)}{9x + 1}$.
Another way: $\frac{16-4x}{9x + 1}=\frac{4(4 - x)}{9x + 1}=\frac{-4(x - 4)}{9x + 1}$. We can also write the numerator as $-4x + 16$ and the fraction is $\frac{-4x + 16}{9x + 1}=\frac{-4(x - 4)}{9x + 1}$.
So the simplified form of the fraction $\frac{8 - x}{9x + 1}-\frac{3x - 8}{9x + 1}$ is $\frac{16 - 4x}{9x + 1}=\frac{4(4 - x)}{9x + 1}$ (or factored form $\frac{-4(x - 4)}{9x + 1}$ or $\frac{4(4 - x)}{9x + 1}$)

Answer:

$\frac{16 - 4x}{9x + 1}$ (or factored form $\frac{-4(x - 4)}{9x + 1}$ or $\frac{4(4 - x)}{9x + 1}$)