Add: 1 over 2(x+4)+1 over 4(4x−2) . Answer should be in mixed number form.

Answer 1

To find the sum of the fractions, we need to first find the least common denominator (LCD) between the two denominators. In this case, the denominators are 2(x+4) and 4(4x-2).

To find the LCD, we need to factor the denominators and identify the highest power of each factor that appears in either fraction.

For 2(x+4), we have one factor of 2.
For 4(4x-2), we have two factors of 2.

So the LCD is 2^2 * (x+4) * (4x-2).

Now, we can rewrite each fraction with the LCD as the denominator:

1 over 2(x+4) = 2(4x-2) / (2^2 * (x+4) * (4x-2))

1 over 4(4x-2) = (x+4) / (2^2 * (x+4) * (4x-2))

Now, we can add the fractions:

2(4x-2) / (2^2 * (x+4) * (4x-2)) + (x+4) / (2^2 * (x+4) * (4x-2))

Now, since the denominators are equal, we can combine the numerators over the common denominator:

(2(4x-2) + (x+4)) / (2^2 * (x+4) * (4x-2))

Simplifying the numerators:

(8x - 4 + x + 4) / (2^2 * (x+4) * (4x-2))

Combining like terms:

(9x) / (2^2 * (x+4) * (4x-2))

Simplifying the denominator:

(9x) / (4 * (x+4) * (4x-2))

Now, since the numerator is already in simplest form, we can write the expression as a mixed number.

The numerator, 9x, cannot be divided by the denominator, 4 * (x+4) * (4x-2), so it remains as it is.

So, the answer in mixed number form is:

9x / (4 * (x+4) * (4x-2))