Question

Which expression is equivalent to
2n/n+4+7/n-1x2n/n+4+7/n-1 if no denominator equals zero?

Answer 1

The expression simplifies to 1 since both the terms on either side of the plus sign simplify down to 1, because they have the same quantities in the numerator and the denominator, thus canceling each other out.

Step-by-step explanation:

The expression given is 2n/(n+4) + 7/(n-1)*2n/(n+4) + 7/(n-1). First, let us simplify the individual fractions. The expression 2n/(n+4) on the left side equals 1 because it has the same quantity in both the numerator and the denominator after simplification, which means they cancel out. Accordingly, any fraction with the same numbers on top and bottom simplifies to 1.

When looking at the right side of the equation, it too simplifies to 1 for the same reason. By maintaining an equality and performing the same operation on both sides of the equals sign, we preserve the balance of the equation. Additional analysis or manipulation is not necessary in this case because both sides reduce to 1, demonstrating that the original expression is equivalent to 1.

Answer 2

The combined fraction is:

`\[ (2n^2 + 5n + 28)/((n + 4)(n - 1)) \]`

This matches option A:

`\[ (2n^2 + 5n + 28)/((n + 4)(n - 1)) \]`

To find which expression is equivalent to
`\( (2n)/(n + 4) + (7)/(n - 1) \)`, we need to combine the two fractions into a single fraction. This requires finding a common denominator, which would be the product of both denominators since they have no common factors (other than 1).

The common denominator will be
`\( (n + 4)(n - 1) \)`. We will adjust each fraction to have this common denominator by multiplying the numerator and denominator of each fraction by the missing factor from the common denominator.

For the first fraction
`\( (2n)/(n + 4) \)`, we need to multiply by
`\( (n - 1)/(n - 1) \)`:

`\[ (2n)/(n + 4) \cdot (n - 1)/(n - 1) = (2n(n - 1))/((n + 4)(n - 1)) \]`

For the second fraction
`\( (7)/(n - 1) \)`, we need to multiply by
`\( (n + 4)/(n + 4) \)`:

`\[ (7)/(n - 1) \cdot (n + 4)/(n + 4) = (7(n + 4))/((n + 4)(n - 1)) \]`

Now we can combine the fractions:

`\[ (2n(n - 1) + 7(n + 4))/((n + 4)(n - 1)) \]`

Expanding the numerators:

`\[ 2n(n - 1) = 2n^2 - 2n \]`

`\[ 7(n + 4) = 7n + 28 \]`

Combine like terms:

`\[ (2n^2 - 2n) + (7n + 28) = 2n^2 + 5n + 28 \]`

The combined fraction is:

`\[ (2n^2 + 5n + 28)/((n + 4)(n - 1)) \]`

This matches option A:

`\[ (2n^2 + 5n + 28)/((n + 4)(n - 1)) \]`

So, the correct answer is A.