Simplify the expression.
$\frac{x+4}{4+x}$
Select the correct choice below and fill in any answer boxes in your choice.
A. $\frac{x+4}{4+x}=$ $\square$
B. The expression cannot be simplified.
Answer (2)
Recognize that x + 4 is equivalent to 4 + x due to the commutative property of addition.
Rewrite the expression as 4 + x 4 + x .
Simplify the fraction by canceling the common factor ( 4 + x ) from the numerator and denominator.
State the simplified expression as 1 .
Explanation
Understanding the Expression We are asked to simplify the expression 4 + x x + 4 .
Using Commutative Property Notice that addition is commutative, which means that x + 4 is the same as 4 + x . Therefore, we can rewrite the expression as 4 + x 4 + x .
Simplifying the Fraction As long as 4 + x is not equal to zero (i.e., x = − 4 ), we can simplify the fraction by canceling the common factor in the numerator and the denominator: 4 + x 4 + x = 1 .
Final Result Therefore, the simplified expression is 1.
Examples
In real life, simplifying expressions like this can be useful when calculating ratios or proportions. For example, if you have a recipe that calls for x + 4 cups of flour and you decide to scale the recipe by a factor of 4 + x x + 4 , you are essentially keeping the recipe the same because the scaling factor simplifies to 1. This ensures that the proportions of all ingredients remain consistent, and the recipe turns out as expected. Understanding how to simplify algebraic expressions helps in maintaining consistency and accuracy in various real-world applications.
The expression 4 + x x + 4 simplifies to 1 because x + 4 is equal to 4 + x . Therefore, the simplified form is 1 as long as x = − 4 .
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