Which expression is equivalent to $\frac{1}{x-2}+\frac{2 x}{5-x}$, for all values of $x$, where $x \neq 2$ and $x \neq 5 ?$
Answer (2)
The equivalent expression to x − 2 1 + 5 − x 2 x is ( x − 2 ) ( 5 − x ) − 2 x 2 + 5 x − 5 . We achieve this by finding a common denominator and simplifying the combined fractions. The final answer is boxed for clarity.
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Rewrite the expression with a common denominator: x − 2 1 − x − 5 2 x .
Combine the fractions: ( x − 2 ) ( x − 5 ) ( x − 5 ) − 2 x ( x − 2 ) .
Simplify the numerator: ( x − 2 ) ( x − 5 ) x − 5 − 2 x 2 + 4 x .
Combine like terms: ( x − 2 ) ( x − 5 ) − 2 x 2 + 5 x − 5 .
( x − 2 ) ( x − 5 ) − 2 x 2 + 5 x − 5
Explanation
Understanding the Problem We are given the expression x − 2 1 + 5 − x 2 x and we want to find an equivalent expression. The conditions x = 2 and x = 5 are also given.
Rewriting the Expression First, rewrite the second fraction to have a common denominator: x − 2 1 + 5 − x 2 x = x − 2 1 − x − 5 2 x
Finding a Common Denominator Now, find a common denominator, which is ( x − 2 ) ( x − 5 ) . Rewrite each fraction with this common denominator: ( x − 2 ) ( x − 5 ) 1 ( x − 5 ) − ( x − 2 ) ( x − 5 ) 2 x ( x − 2 )
Combining the Fractions Combine the fractions: ( x − 2 ) ( x − 5 ) ( x − 5 ) − 2 x ( x − 2 )
Simplifying the Numerator Simplify the numerator: ( x − 2 ) ( x − 5 ) x − 5 − 2 x 2 + 4 x
Combining Like Terms Combine like terms in the numerator: ( x − 2 ) ( x − 5 ) − 2 x 2 + 5 x − 5
Final Answer The equivalent expression is ( x − 2 ) ( x − 5 ) − 2 x 2 + 5 x − 5 .
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.
