Add or subtract.
$\frac{7-x}{x-3}-\frac{2 x-5}{3-x}$
A. $-\frac{x+2}{x-3}$
B. $\frac{x+2}{x-3}$
C. $\frac{x+12}{x-3}$
D. $-\frac{x+12}{x-3}$
Answer (1)
Rewrite the second fraction with the denominator x − 3 .
Combine the fractions.
Simplify the expression.
The correct answer is x − 3 x + 2 .
Explanation
Understanding the Problem We are asked to simplify the expression x − 3 7 − x − 3 − x 2 x − 5 and choose the correct answer from the list of possible answers.
Finding a Common Denominator To simplify the expression, we need to have a common denominator. Notice that 3 − x = − ( x − 3 ) . Therefore, we can rewrite the second fraction as follows: 3 − x 2 x − 5 = − ( x − 3 ) 2 x − 5 = − x − 3 2 x − 5 Now we can rewrite the original expression as: x − 3 7 − x − 3 − x 2 x − 5 = x − 3 7 − x − ( − x − 3 2 x − 5 ) = x − 3 7 − x + x − 3 2 x − 5 Now that we have a common denominator, we can combine the fractions.
Combining the Fractions Combining the fractions, we get: x − 3 7 − x + 2 x − 5 = x − 3 x + 2 So the simplified expression is x − 3 x + 2 .
Choosing the Correct Answer Comparing our simplified expression with the given options, we see that the correct answer is x − 3 x + 2 .
Examples
Rational expressions are used in many areas of science and engineering to model relationships between different quantities. For example, in physics, they can be used to describe the motion of objects or the behavior of electrical circuits. In economics, they can be used to model supply and demand curves. Simplifying rational expressions allows us to better understand these relationships and make predictions about the systems they describe.
