Simplify.
$\frac{3}{x-3}-\frac{5}{x+2}$

Find a common denominator: ( x − 3 ) ( x + 2 ) .
Rewrite each fraction with the common denominator.
Combine and simplify the numerators: 3 ( x + 2 ) − 5 ( x − 3 ) = − 2 x + 21 .
The simplified expression is x 2 − x − 6 − 2 x + 21 ​ ​ .

Explanation

Understanding the Problem We are given the expression x − 3 3 ​ − x + 2 5 ​ . Our goal is to simplify this expression by combining the two fractions into a single fraction.

Finding a Common Denominator To combine the two fractions, we need to find a common denominator. The common denominator for the two fractions is ( x − 3 ) ( x + 2 ) .

Rewriting Fractions with Common Denominator Now, we rewrite each fraction with the common denominator: x − 3 3 ​ − x + 2 5 ​ = ( x − 3 ) ( x + 2 ) 3 ( x + 2 ) ​ − ( x − 3 ) ( x + 2 ) 5 ( x − 3 ) ​

Combining Numerators Next, we combine the numerators: ( x − 3 ) ( x + 2 ) 3 ( x + 2 ) − 5 ( x − 3 ) ​

Expanding the Numerator Now, we expand the numerator: ( x − 3 ) ( x + 2 ) 3 x + 6 − 5 x + 15 ​

Simplifying the Numerator Then, we simplify the numerator: ( x − 3 ) ( x + 2 ) − 2 x + 21 ​

Expanding the Denominator We can also expand the denominator to get: x 2 − 3 x + 2 x − 6 − 2 x + 21 ​ = x 2 − x − 6 − 2 x + 21 ​

Final Simplified Expression Therefore, the simplified expression is: x 2 − x − 6 − 2 x + 21 ​


Examples
When dealing with rates of work or flow that are inversely proportional to some variable, you often encounter expressions like the one we just simplified. For instance, if one pipe fills a tank at a rate of x − 3 3 ​ tanks per hour and another empties the same tank at a rate of x + 2 5 ​ tanks per hour, the combined rate can be found by subtracting the two fractions, leading to the simplified expression. This type of problem is also applicable in scenarios involving electrical circuits, mixing solutions, and comparing investment returns.

Answered by GinnyAnswer | 2025-07-07