\frac{e-2}{e+2}-\frac{e-1}{e+3}
Answer (1)
Find a common denominator: ( e + 2 ) ( e + 3 ) .
Rewrite the fractions with the common denominator and expand the numerators: ( e + 2 ) ( e + 3 ) ( e 2 + e − 6 ) − ( e 2 + e − 2 ) .
Simplify the numerator by combining like terms: ( e + 2 ) ( e + 3 ) − 4 .
The simplified expression is: e 2 + 5 e + 6 − 4 .
Explanation
Understanding the Problem We are asked to simplify the expression e + 2 e − 2 − e + 3 e − 1 . To do this, we need to find a common denominator and combine the fractions.
Finding a Common Denominator The common denominator for the two fractions is ( e + 2 ) ( e + 3 ) . We rewrite each fraction with this common denominator: e + 2 e − 2 − e + 3 e − 1 = ( e + 2 ) ( e + 3 ) ( e − 2 ) ( e + 3 ) − ( e + 2 ) ( e + 3 ) ( e − 1 ) ( e + 2 )
Expanding the Numerators Now, we expand the numerators: ( e + 2 ) ( e + 3 ) ( e 2 + 3 e − 2 e − 6 ) − ( e 2 + 2 e − e − 2 ) = ( e + 2 ) ( e + 3 ) ( e 2 + e − 6 ) − ( e 2 + e − 2 )
Combining the Fractions Next, we combine the numerators: ( e + 2 ) ( e + 3 ) e 2 + e − 6 − e 2 − e + 2 = ( e + 2 ) ( e + 3 ) − 4
Simplifying the Expression Finally, we can rewrite the denominator in expanded form: e 2 + 3 e + 2 e + 6 − 4 = e 2 + 5 e + 6 − 4 Thus, the simplified expression is e 2 + 5 e + 6 − 4 .
Examples
Rational expressions are useful in many areas of science and engineering. For example, in physics, they can be used to describe the relationship between voltage, current, and resistance in an electrical circuit. In economics, they can be used to model cost and revenue functions. Simplifying rational expressions makes it easier to analyze and understand these relationships.
