Question
1. Which expression is equivalent to the one shown below?
(3x + 16)/(6x + 36)
A.2
b.1/2
c.3/4
D 1/3
(3x + 16)/(6x + 36)
A.2
b.1/2
c.3/4
D 1/3
Answers
Answered by
GPT-4o mini
To simplify the expression \((3x + 16)/(6x + 36)\), we can factor both the numerator and the denominator.
1. In the numerator \( 3x + 16 \) cannot be simplified further.
2. In the denominator \( 6x + 36\), we can factor out a common factor of 6:
\[
6x + 36 = 6(x + 6)
\]
So the expression becomes:
\[
\frac{3x + 16}{6(x + 6)}
\]
This expression doesn't simplify further in terms of cancellation between the numerator and denominator since there are no common factors.
Now, to evaluate how the entire expression behaves when \( x \) approaches specific values is helpful, but we can't determine a numerical constant from simplification alone that would correspond to one of the options (A, B, C, D).
Thus, let's substitute a particular value for \( x \) to find the corresponding numeric output:
1. If we let \( x = 0 \):
\[
\frac{3(0) + 16}{6(0) + 36} = \frac{16}{36} = \frac{4}{9}
\]
2. If we let \( x = 1 \):
\[
\frac{3(1) + 16}{6(1) + 36} = \frac{3 + 16}{6 + 36} = \frac{19}{42}
\]
None of these evaluations lead to a clear-cut answer matching A, B, C, or D.
Let’s check if there are point values for which the original expression results in a value of A, B, C, or D:
To verify the specific values of the options, we can check values that approach certain numbers:
1. Calculating for \( x = -6 \) which leads to a denominator of zero:
- This signifies the limit must be evaluated for x approaching -6 instead, and the concept of simplifying shows down to certain constants.
Ultimately, only simplistically or through calculated limits will provide insights, where \( \frac{4}{9} \) and \( \frac{19}{42} \) respectively become less apparent.
Given the complexity, if you just need an answer based on the problem: the options look to indicate reductions or ratios on bounds; yet without further confines on \( x \) it’s wise to numerically solve instead.
### In conclusive terms:
No single simplification led directly to outputs within the boxed (A) to (D) options; thus unless there is an oversight in perception numerically or algebraically, resulting calculations clearly relay mixed fractions approaching 4/9 or approximations, needing limits for functions for more pointed numerical identification.
If there is a specific anticipated value or examination of bounds instead of outright numeric value reductions, please clarify where further numerical approaches should emphasize.
1. In the numerator \( 3x + 16 \) cannot be simplified further.
2. In the denominator \( 6x + 36\), we can factor out a common factor of 6:
\[
6x + 36 = 6(x + 6)
\]
So the expression becomes:
\[
\frac{3x + 16}{6(x + 6)}
\]
This expression doesn't simplify further in terms of cancellation between the numerator and denominator since there are no common factors.
Now, to evaluate how the entire expression behaves when \( x \) approaches specific values is helpful, but we can't determine a numerical constant from simplification alone that would correspond to one of the options (A, B, C, D).
Thus, let's substitute a particular value for \( x \) to find the corresponding numeric output:
1. If we let \( x = 0 \):
\[
\frac{3(0) + 16}{6(0) + 36} = \frac{16}{36} = \frac{4}{9}
\]
2. If we let \( x = 1 \):
\[
\frac{3(1) + 16}{6(1) + 36} = \frac{3 + 16}{6 + 36} = \frac{19}{42}
\]
None of these evaluations lead to a clear-cut answer matching A, B, C, or D.
Let’s check if there are point values for which the original expression results in a value of A, B, C, or D:
To verify the specific values of the options, we can check values that approach certain numbers:
1. Calculating for \( x = -6 \) which leads to a denominator of zero:
- This signifies the limit must be evaluated for x approaching -6 instead, and the concept of simplifying shows down to certain constants.
Ultimately, only simplistically or through calculated limits will provide insights, where \( \frac{4}{9} \) and \( \frac{19}{42} \) respectively become less apparent.
Given the complexity, if you just need an answer based on the problem: the options look to indicate reductions or ratios on bounds; yet without further confines on \( x \) it’s wise to numerically solve instead.
### In conclusive terms:
No single simplification led directly to outputs within the boxed (A) to (D) options; thus unless there is an oversight in perception numerically or algebraically, resulting calculations clearly relay mixed fractions approaching 4/9 or approximations, needing limits for functions for more pointed numerical identification.
If there is a specific anticipated value or examination of bounds instead of outright numeric value reductions, please clarify where further numerical approaches should emphasize.