What is the difference?
$\frac{x}{x^2-16}-\frac{3}{x-4}$
$\frac{2(x+6)}{(x+4)(x-4)}$
$\frac{-2(x+6)}{(x+4)(x-4)}$
$\frac{x-3}{(x+5)(x-4)}$
$\frac{-2(x-6)}{(x+4)(x-4)}$
Answer (1)
Simplify the first expression by finding a common denominator and combining terms.
Factor the numerator of the simplified expression.
Compare the simplified expression to the other given expressions.
Identify the expression that is equivalent to the simplified first expression: ( x + 4 ) ( x − 4 ) − 2 ( x + 6 ) .
Explanation
Understanding the Problem We are given five expressions and asked to find 'the difference'. This is ambiguous, so I will assume that the question is asking us to simplify the first expression and then determine which, if any, of the other four expressions are equivalent to the simplified first expression.
Simplifying Expression 1 Let's simplify the first expression: x 2 − 16 x − x − 4 3
Factoring the Denominator First, we factor the denominator of the first term: ( x − 4 ) ( x + 4 ) x − x − 4 3
Finding a Common Denominator Next, we find a common denominator: ( x − 4 ) ( x + 4 ) x − ( x − 4 ) ( x + 4 ) 3 ( x + 4 )
Combining Fractions Now, we combine the fractions: ( x − 4 ) ( x + 4 ) x − 3 ( x + 4 )
Simplifying the Numerator We simplify the numerator: ( x − 4 ) ( x + 4 ) x − 3 x − 12 = ( x − 4 ) ( x + 4 ) − 2 x − 12
Factoring the Numerator We factor out a -2 from the numerator: ( x − 4 ) ( x + 4 ) − 2 ( x + 6 )
Comparing Expressions Now, we compare the simplified expression 1, which is ( x − 4 ) ( x + 4 ) − 2 ( x + 6 ) , to expressions 2, 3, 4, and 5.
Identifying Equivalent Expressions Expression 2: ( x + 4 ) ( x − 4 ) 2 ( x + 6 ) Expression 3: ( x + 4 ) ( x − 4 ) − 2 ( x + 6 ) Expression 4: ( x + 5 ) ( x − 4 ) x − 3 Expression 5: ( x + 4 ) ( x − 4 ) − 2 ( x − 6 ) We can see that expression 3, ( x + 4 ) ( x − 4 ) − 2 ( x + 6 ) , is equivalent to the simplified expression 1.
Final Answer Therefore, the simplified form of the first expression is equal to expression 3.
Examples
Simplifying rational expressions is a fundamental skill in algebra and is used extensively in calculus and other higher-level math courses. For example, when solving for the area between two curves, you may need to simplify complex rational expressions resulting from integration. Also, in physics, simplifying rational expressions can help in analyzing circuits or understanding wave behavior. By mastering these techniques, you can tackle more complex problems in various fields.
