Problem 99 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The rational expressions $$\frac{7}{14 x} \text { and } \frac{7}{14+x}$$ can both be simplified by dividing each numerator and each denominator by 7

Step-by-Step Solution

Answer

The statement does not make sense, because while we can simplify \( \frac{7}{14x} \) by dividing the numerator and the denominator by 7, we cannot simplify the expression \( \frac{7}{14+x} \) in the same way.

Step 1: Analyze the first expression

The first given expression is \( \frac{7}{14x} \). This expression can indeed be simplified by dividing the numerator and the denominator by 7. This is because the term 14x is composed by multiplying 14 and x, i.e., it's in the form of multiplication.

Step 2: Simplify the first expression

Dividing both the numerator and denominator by 7 for first expression gives \( \frac{7/7}{(14*7)/7} = \frac{1}{2x} \). Therefore, the statement makes sense for the first expression.

Step 3: Analyze the second expression

The second given expression is \( \frac{7}{14+x} \). In this case, the denominator is composed by adding 14 and x. If we try to simplify this by dividing each numerator and denominator by 7, we do not get a simpler expression because the operation in the denominator is addition, not multiplication.

Step 4: Try to simplify the second expression

Attempting to divide both the numerator and denominator by 7 for the second expression gives \( \frac{7/7}{(14/7)+(x/7)} = \frac{1}{2+(x/7)} \). This expression is not simpler than the original one. Therefore, the statement does not make sense for the second expression.

Step 5: Final comparison and conclusion

The statement affirms that both expressions can be simplified by dividing each numerator and each denominator by 7. The analysis shows that this is untrue for the second expression, \( \frac{7}{14+x} \), thus the statement does not make sense.