Question
Express in simplest form: 3x+18/4x+8÷x2−36/x+2 .(1 point) Responses 4(x−6)3 Start Fraction 4 left parenthesis x minus 6 right parenthesis over 3 End Fraction 34x−24 Start Fraction 2 over 4 x minus 24 End Fraction 34x−8 Start Fraction 3 over 4 x minus 8 End Fraction 14(x−2)
Answers
GPT-4o mini
To simplify the expression \(\frac{3x + 18}{4x + 8} \div \frac{x^2 - 36}{x + 2}\), we will first rewrite the division as multiplication by the reciprocal.
1. Rewrite the expression:
\[
\frac{3x + 18}{4x + 8} \times \frac{x + 2}{x^2 - 36}
\]
2. Factor where possible:
- The numerator \(3x + 18\) can be factored as:
\[
3(x + 6)
\]
- The denominator \(4x + 8\) can be factored as:
\[
4(x + 2)
\]
- The term \(x^2 - 36\) is a difference of squares, which can be factored as:
\[
(x - 6)(x + 6)
\]
3. Substitute the factored forms back into the expression:
\[
\frac{3(x + 6)}{4(x + 2)} \times \frac{x + 2}{(x - 6)(x + 6)}
\]
4. Now, simplify by canceling common factors:
- \(x + 2\) cancels in the numerator and denominator:
\[
\frac{3(x + 6)}{4} \times \frac{1}{(x - 6)(x + 6)}
\]
- The \(x + 6\) in the numerator and denominator also cancels:
\[
\frac{3}{4(x - 6)}
\]
5. The simplest form of the expression is:
\[
\frac{3}{4(x - 6)}
\]
However, from the available responses, it seems I made a mistake in the path of simplification or overlooked the forms offered in your options. The simplified version corresponds to rearranging terms:
\(\frac{3}{4(x - 6)}\) is simplified out, but the form \(4(x - 6)^{-1}\) or similar might resemble in your options.
If we check closely, your provided options reflect as:
\(\frac{3}{4(x - 6)}\), which isn't listed, implying they indeed might have alternatives or corrections.
Among the choices, if I were to verify if \(4(x - 6)^{-1}\) could match as well—´recheck whether likely transforming occurs. Still, \(\frac{3}{4(x - 6)}\) would work as a basis.
Final response not listed among prominent answers but mathematically verified yields:
\(\frac{3}{4(x - 6)}\).
If "responses" imply recheck on any missed distinctions to verify contestant outputs.
1. Rewrite the expression:
\[
\frac{3x + 18}{4x + 8} \times \frac{x + 2}{x^2 - 36}
\]
2. Factor where possible:
- The numerator \(3x + 18\) can be factored as:
\[
3(x + 6)
\]
- The denominator \(4x + 8\) can be factored as:
\[
4(x + 2)
\]
- The term \(x^2 - 36\) is a difference of squares, which can be factored as:
\[
(x - 6)(x + 6)
\]
3. Substitute the factored forms back into the expression:
\[
\frac{3(x + 6)}{4(x + 2)} \times \frac{x + 2}{(x - 6)(x + 6)}
\]
4. Now, simplify by canceling common factors:
- \(x + 2\) cancels in the numerator and denominator:
\[
\frac{3(x + 6)}{4} \times \frac{1}{(x - 6)(x + 6)}
\]
- The \(x + 6\) in the numerator and denominator also cancels:
\[
\frac{3}{4(x - 6)}
\]
5. The simplest form of the expression is:
\[
\frac{3}{4(x - 6)}
\]
However, from the available responses, it seems I made a mistake in the path of simplification or overlooked the forms offered in your options. The simplified version corresponds to rearranging terms:
\(\frac{3}{4(x - 6)}\) is simplified out, but the form \(4(x - 6)^{-1}\) or similar might resemble in your options.
If we check closely, your provided options reflect as:
\(\frac{3}{4(x - 6)}\), which isn't listed, implying they indeed might have alternatives or corrections.
Among the choices, if I were to verify if \(4(x - 6)^{-1}\) could match as well—´recheck whether likely transforming occurs. Still, \(\frac{3}{4(x - 6)}\) would work as a basis.
Final response not listed among prominent answers but mathematically verified yields:
\(\frac{3}{4(x - 6)}\).
If "responses" imply recheck on any missed distinctions to verify contestant outputs.