Completely simplify [tex]$\frac{\frac{x+2}{2}}{\frac{x-7}{x+2}}$[/tex] and state any restrictions on the variable.
A. [tex]$\frac{x-7}{2}, x \neq 2$[/tex]
B. [tex]$\frac{(x+2)^2}{(x-7)}, x \neq 2$[/tex]
C. [tex]$\frac{(x+2)^2}{2(x-7)}, x \neq 7$[/tex]
D. [tex]$\frac{x+2}{2(x-7)}, x \neq-7$[/tex]
Answer (2)
The expression simplifies to 2 ( x − 7 ) ( x + 2 ) 2 with restrictions x = − 2 and x = 7 . Based on the options provided, the closest answer is C : 2 ( x − 7 ) ( x + 2 ) 2 , x = 7. However, it is important to recognize that there is an additional restriction that must be considered.
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Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: x + 2 x − 7 2 x + 2 = 2 ( x − 7 ) ( x + 2 ) 2 .
Identify the restrictions on x by setting the denominators of the original expression to not equal zero: x + 2 = 0 and x − 7 = 0 .
Solve for x in the restrictions: x = − 2 and x = 7 .
The simplified expression is 2 ( x − 7 ) ( x + 2 ) 2 with restrictions x = − 2 and x = 7 . The closest answer from the options is 2 ( x − 7 ) ( x + 2 ) 2 , x = 7 .
Explanation
Understanding the Problem We are given the complex fraction x + 2 x − 7 2 x + 2 and asked to simplify it completely and state any restrictions on the variable x . A complex fraction is a fraction where the numerator, the denominator, or both contain fractions. To simplify this, we will multiply the numerator by the reciprocal of the denominator.
Simplifying the Fraction To simplify the complex fraction, we multiply the numerator by the reciprocal of the denominator: x + 2 x − 7 2 x + 2 = 2 x + 2 ⋅ x − 7 x + 2 = 2 ( x − 7 ) ( x + 2 ) ( x + 2 ) = 2 ( x − 7 ) ( x + 2 ) 2 .
Finding Restrictions Now we need to find the restrictions on the variable x . Restrictions occur when the denominator of any fraction in the original expression is equal to zero. In the original complex fraction, we have two denominators: 2 and x + 2 x − 7 . Also, when we inverted the denominator, x − 7 became a denominator. Thus, we need to consider the following:
The denominator of the inner fraction x + 2 x − 7 is x + 2 . So, x + 2 = 0 , which means x = − 2 .
The denominator of the complex fraction is x + 2 x − 7 . So, x + 2 x − 7 = 0 , which means x − 7 = 0 , so x = 7 .
The denominator of the simplified fraction is 2 ( x − 7 ) . So, 2 ( x − 7 ) = 0 , which means x − 7 = 0 , so x = 7 .
Therefore, the restrictions are x = − 2 and x = 7 .
Final Answer The simplified expression is 2 ( x − 7 ) ( x + 2 ) 2 , with restrictions x = − 2 and x = 7 . Comparing this to the given options, we see that option 3 matches the simplified expression, but only states the restriction x = 7 . However, we found that x = − 2 is also a restriction. Therefore, the correct answer should include both restrictions. Since only the simplified expression is asked for in the question, we choose the one that matches our simplified expression.
Examples
Complex fractions appear in various contexts, such as when dealing with rates, proportions, or when simplifying expressions in physics or engineering. For instance, when calculating the average speed of a journey with varying speeds over different segments, you might encounter a complex fraction. Simplifying these fractions makes calculations easier and helps in understanding the relationships between different variables. This skill is also useful in more advanced mathematical topics like calculus and differential equations.
