Simplify the following rational expressions: [tex]$\frac{4}{x+2}-\frac{3}{x-1}$[/tex]
Answer (2)
∙ Find a common denominator for the two fractions: ( x + 2 ) ( x − 1 ) .
∙ Rewrite the expression with the common denominator: ( x + 2 ) ( x − 1 ) 4 ( x − 1 ) − 3 ( x + 2 ) .
∙ Simplify the numerator: 4 ( x − 1 ) − 3 ( x + 2 ) = x − 10 .
∙ The simplified expression is ( x + 2 ) ( x − 1 ) x − 10 .
Explanation
Problem Analysis We are asked to simplify the expression x + 2 4 − x − 1 3 and determine which of the given expressions it is equal to.
Finding Common Denominator To simplify the expression, we need to find a common denominator for the two fractions. The common denominator is ( x + 2 ) ( x − 1 ) .
Rewriting with Common Denominator Now, we rewrite the expression with the common denominator: x + 2 4 − x − 1 3 = ( x + 2 ) ( x − 1 ) 4 ( x − 1 ) − ( x + 2 ) ( x − 1 ) 3 ( x + 2 )
Combining Fractions Combine the fractions: ( x + 2 ) ( x − 1 ) 4 ( x − 1 ) − 3 ( x + 2 )
Simplifying Numerator Expand and simplify the numerator: 4 ( x − 1 ) − 3 ( x + 2 ) = 4 x − 4 − 3 x − 6 = x − 10
Simplified Expression So, the simplified expression is: ( x + 2 ) ( x − 1 ) x − 10
Final Answer Comparing this to the given expressions, we see that it matches the second expression: ( x + 2 ) ( x − 1 ) x − 10
Examples
Rational expressions are useful in various fields, such as physics and engineering, where they are used to model relationships between different variables. For example, in electrical engineering, rational expressions can be used to describe the impedance of a circuit as a function of frequency. Simplifying these expressions allows engineers to analyze and design circuits more efficiently. Similarly, in physics, rational expressions can appear in equations describing the motion of objects or the behavior of waves. Simplifying these expressions can make it easier to solve these equations and understand the underlying physical phenomena. Understanding how to manipulate and simplify rational expressions is a fundamental skill in many scientific and technical disciplines.
To simplify x + 2 4 − x − 1 3 , we find a common denominator, rewrite each fraction, combine them, and then simplify the numerator. The final simplified expression is ( x + 2 ) ( x − 1 ) x − 10 .
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