Question

Subtract  x3x+7−x9x+21 .(1 point)Responses2x9x+21Start Fraction 2 x over 9 x plus 21 End Fraction4x9x+27Start Fraction 4 x over 9 x plus 27 End Fraction00x3x+7Start Fraction x over 3 x plus 7 End Fraction

Answer 1

To subtract the expression \( \frac{x}{3x+7} - \frac{x}{9x+21} \), we will first find a common denominator and then perform the subtraction.

### Step 1: Identify the denominators
The denominators are:
1. \( 3x + 7 \)
2. \( 9x + 21 \)

We can factor \( 9x + 21 \) as:
\[
9x + 21 = 9(x + \frac{7}{3}) = 3(3x + 7)
\]
However, the common denominator for the two fractions is the product of the distinct linear factors:
\[
(3x + 7)(9x + 21) = (3x + 7)(3(3x + 7)) = 3(3x + 7)^2
\]

### Step 2: Rewrite each term with the common denominator
Now, rewrite each fraction:
1. \( \frac{x}{3x+7} \) must be multiplied by \( \frac{9x+21}{9x+21} \)
2. \( \frac{x}{9x+21} \) must be multiplied by \( \frac{3x+7}{3x+7} \)

### Step 3: Rewrite the expressions
The first term becomes:
\[
\frac{x(9x + 21)}{(3x + 7)(9x + 21)} = \frac{9x^2 + 21x}{(3x + 7)(9x + 21)}
\]
The second term becomes:
\[
\frac{x(3x + 7)}{(3x + 7)(9x + 21)} = \frac{3x^2 + 7x}{(3x + 7)(9x + 21)}
\]

### Step 4: Subtract the two fractions
Now we can subtract them:
\[
\frac{9x^2 + 21x - (3x^2 + 7x)}{(3x + 7)(9x + 21)} = \frac{(9x^2 - 3x^2) + (21x - 7x)}{(3x + 7)(9x + 21)} = \frac{6x^2 + 14x}{(3x + 7)(9x + 21)}
\]

### Step 5: Factor the numerator
The numerator can be factored:
\[
6x^2 + 14x = 2x(3x + 7)
\]
Putting this back into our expression gives:
\[
\frac{2x(3x + 7)}{(3x + 7)(9x + 21)}
\]

### Step 6: Cancel common factors
Since \( 3x + 7 \) appears in both the numerator and the denominator, we can cancel it (assuming \( 3x + 7 \neq 0 \)):
\[
\frac{2x}{9x + 21}
\]

### Final Result
The result is:
\[
\frac{2x}{9x + 21}
\]

So from the given options, the answer is:
\[
\boxed{\frac{2x}{9x + 21}}
\]