Which of the following is equivalent to [tex]$\frac{1}{x+2}-\frac{1}{x-3}$[/tex]?
Answer (2)
The expression x + 2 1 − x − 3 1 simplifies to ( x + 2 ) ( x − 3 ) − 5 . This was done by finding a common denominator, rewriting each fraction, subtracting, and simplifying. The final equivalent expression is ( x + 2 ) ( x − 3 ) − 5 .
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Find a common denominator: ( x + 2 ) ( x − 3 ) .
Rewrite the fractions with the common denominator.
Subtract the fractions: ( x + 2 ) ( x − 3 ) x − 3 − ( x + 2 ) ( x − 3 ) x + 2 = ( x + 2 ) ( x − 3 ) ( x − 3 ) − ( x + 2 ) .
Simplify the expression: ( x + 2 ) ( x − 3 ) − 5 .
The equivalent expression is ( x + 2 ) ( x − 3 ) − 5 .
Explanation
Understanding the Problem We are given the expression x + 2 1 − x − 3 1 and we want to find an equivalent expression. To do this, we will combine the two fractions into a single fraction.
Finding a Common Denominator To combine the fractions, we need a common denominator. The common denominator is ( x + 2 ) ( x − 3 ) . We rewrite each fraction with this denominator: x + 2 1 = ( x + 2 ) ( x − 3 ) 1 ( x − 3 ) = ( x + 2 ) ( x − 3 ) x − 3 x − 3 1 = ( x − 3 ) ( x + 2 ) 1 ( x + 2 ) = ( x + 2 ) ( x − 3 ) x + 2
Subtracting the Fractions Now we can subtract the two fractions: x + 2 1 − x − 3 1 = ( x + 2 ) ( x − 3 ) x − 3 − ( x + 2 ) ( x − 3 ) x + 2 = ( x + 2 ) ( x − 3 ) ( x − 3 ) − ( x + 2 ) = ( x + 2 ) ( x − 3 ) x − 3 − x − 2 = ( x + 2 ) ( x − 3 ) − 5
Final Answer So, the equivalent expression is ( x + 2 ) ( x − 3 ) − 5 .
Examples
This type of algebraic manipulation is useful in calculus when integrating rational functions using partial fraction decomposition. For example, if you have a complicated rational function, you can break it down into simpler fractions, making it easier to integrate. This technique is also used in circuit analysis and control systems to simplify transfer functions.
