Given that 𝑥 is greater than four, which of the following is equivalent to two divided by one over 𝑥 minus one plus one over 𝑥 plus four? Is it A) two 𝑥 squared plus six 𝑥 minus eight? B) Two 𝑥 squared plus six 𝑥 minus eight divided by two 𝑥 plus three? C) Two 𝑥 plus three? Or D) two 𝑥 plus three divided by two 𝑥 squared plus six 𝑥 minus eight?
In order to find the equivalent fraction, we firstly need to simplify the denominator. The denominator wants us to add one over 𝑥 minus one plus one over 𝑥 plus four. In order to add or subtract two algebraic fractions, we firstly have to find a common denominator. The easiest way to do this is to multiply the two denominators together. In this case, we need to multiply 𝑥 minus one by 𝑥 plus four.
As we have multiplied the bottom of the first fraction by 𝑥 plus four, we need to do the same to the top. We have multiplied the bottom of the second fraction by 𝑥 minus one, so we need to multiply the top by 𝑥 minus one. This is often called cross multiplying and leaves us with one multiplied by 𝑥 plus four plus one multiplied by 𝑥 minus one. We now need to distribute the parentheses on the top and bottom of the fraction. One multiplied by 𝑥 is equal to one 𝑥, or 𝑥. And one multiplied by four is equal to four. The second bracket, or parentheses, on the numerator expands to give us 𝑥 minus one.
We can distribute the parentheses on the denominator by using the FOIL method. Multiplying the first terms gives us 𝑥 squared, as 𝑥 multiplied by 𝑥 is 𝑥 squared. Multiplying the outside terms gives us positive four 𝑥. Multiplying the inside terms gives us negative one 𝑥, or negative 𝑥. And finally, multiplying the last terms gives us negative four. Multiplying a negative number by a positive number gives us a negative answer.
We can simplify the numerator by collecting like terms. 𝑥 plus 𝑥 is equal to two 𝑥. And four minus one is equal to three. The denominator simplifies to 𝑥 squared plus three 𝑥 minus four, as four 𝑥 minus 𝑥 is equal to three 𝑥. This means that one over 𝑥 minus one plus one over 𝑥 plus four is equivalent to two 𝑥 plus three over 𝑥 squared plus three 𝑥 minus four. Looking back to our original expression, we need to divide two by one over 𝑥 minus one plus one over 𝑥 plus four. This means we need to divide two by two 𝑥 plus three over 𝑥 squared plus three 𝑥 minus four.
Dividing by a fraction is the same as multiplying by the reciprocal of the fraction. The reciprocal means one divided by the number. This means that the numerator becomes the denominator, and vice versa. We need to multiply two by 𝑥 squared plus three 𝑥 minus four over two 𝑥 plus three. One way of remembering this is K C F. We keep the first number the same. We change the sign from a division to a multiplication. And we flip the fraction that we are dividing by.
The number two can be written as two over one, or two divided by one. We then need to multiply the two denominators and the two numerators. Two multiplied by 𝑥 squared plus three 𝑥 minus four is equal to two 𝑥 squared plus six 𝑥 minus eight. And one multiplied by two 𝑥 plus three is equal to two 𝑥 plus three. The expression two divided by one over 𝑥 minus one plus one over 𝑥 plus four is equivalent to two 𝑥 squared plus six 𝑥 minus eight divided by two 𝑥 plus three. The correct option was option B.