Questions: Subtract. Simplify your answer. [ frac4 x16 x^2-9-frac43-4 x= ]

$Subtract. Simplify your answer. [ frac4 x16 x^2-9-frac43-4 x= ]$

Transcript text: Subtract. Simplify your answer. \[ \frac{4 x}{16 x^{2}-9}-\frac{4}{3-4 x}= \] $\square$

Solution

Solution Steps

To subtract these fractions, we need to find a common denominator. First, factor the denominators if possible. Then, rewrite each fraction with the common denominator and perform the subtraction. Finally, simplify the resulting fraction if possible.

Step 1: Factor the Denominators

First, we factor the denominators of the given fractions: \[ \frac{4x}{16x^2 - 9} - \frac{4}{3 - 4x} \] The denominator $16x^2 - 9$ can be factored as a difference of squares: \[ 16x^2 - 9 = (4x + 3)(4x - 3) \]

Step 2: Rewrite the Fractions with a Common Denominator

Next, we rewrite each fraction with the common denominator $(4x + 3)(4x - 3)$: \[ \frac{4x}{(4x + 3)(4x - 3)} - \frac{4}{3 - 4x} \] Notice that $3 - 4x$ can be rewritten as $-(4x - 3)$. Thus, the second fraction becomes: \[ \frac{4}{-(4x - 3)} = -\frac{4}{4x - 3} \] Rewriting the second fraction with the common denominator: \[ -\frac{4(4x + 3)}{(4x + 3)(4x - 3)} \]

Step 3: Perform the Subtraction

Now, we perform the subtraction: \[ \frac{4x}{(4x + 3)(4x - 3)} - \frac{4(4x + 3)}{(4x + 3)(4x - 3)} = \frac{4x - 4(4x + 3)}{(4x + 3)(4x - 3)} \] Simplify the numerator: \[ 4x - 4(4x + 3) = 4x - 16x - 12 = -12 - 12x \] Thus, the expression becomes: \[ \frac{-12 - 12x}{(4x + 3)(4x - 3)} \]

Step 4: Simplify the Result

Factor out $-12$ from the numerator: \[ \frac{-12(1 + x)}{(4x + 3)(4x - 3)} \]