Final answer:
To simplify the expression 3x/(x+7) + 4/(2x-1), find a common denominator, adjust each fraction to have this common denominator, then combine and simplify the resulting numerators over the common denominator.
Step-by-step explanation:
To simplify the expression 3x/(x+7) + 4/(2x-1) into a single fraction, we need to find a common denominator. The denominators are (x+7) and (2x-1), so the common denominator will be the product of these two, which is (x+7)(2x-1).
Next, we write each fraction with the common denominator by multiplying the numerator and denominator of each fraction by whatever is missing from its current denominator to match the common denominator:
- For the first fraction, we multiply both numerator and denominator by (2x-1), resulting in (3x(2x-1))/((x+7)(2x-1)).
- For the second fraction, we multiply both numerator and denominator by (x+7), giving us (4(x+7))/((x+7)(2x-1)).
We then add the two fractions:
(3x(2x-1) + 4(x+7))/((x+7)(2x-1))
Now, expand and combine the numerators:
(6x^2 - 3x + 4x + 28)/((x+7)(2x-1))
Simplify the numerator, combining like terms:
(6x^2 + x + 28)/((x+7)(2x-1))
Now, we have a single fraction representing the simplified form of the expression.