The Correct Answer and Explanation is:
Of course. Here is the correct answer and a detailed explanation.
Correct Answer: 5
Explanation
The problem asks us to solve the rational equation 1/x + 2/(3x) = 1/3. Solving a rational equation involves a few key steps to find the value of x that makes the statement true.
First, it is important to identify any values of x that are not allowed. These are known as excluded values, and they occur when a denominator equals zero, as division by zero is undefined. In this equation, the denominators are x and 3x. If x were 0, both denominators would be zero. Therefore, x cannot equal 0. We must remember this to check if our final answer is an extraneous solution.
The next step is to eliminate the fractions to make the equation easier to solve. We can do this by finding the least common denominator (LCD) of all the fractions. The denominators are x, 3x, and 3. The smallest expression that all three denominators divide into evenly is 3x.
Now, we multiply every term in the entire equation by the LCD, which is 3x:
3x * (1/x) + 3x * (2/(3x)) = 3x * (1/3)
This multiplication allows us to cancel the denominators. In the first term, the x in 3x cancels with the x in the denominator, leaving 3 * 1, or 3. In the second term, the entire 3x cancels with the denominator, leaving just 2. On the right side of the equation, the 3 in 3x cancels with the denominator, leaving x * 1, or x.
After simplifying, we are left with a much simpler linear equation:
3 + 2 = x
Solving for x is now straightforward:
5 = x
Finally, we must check if this solution is valid. Our only excluded value was x = 0. Since our solution is x = 5, which is not equal to 0, the solution is valid and not extraneous. The final answer is 5.
