Simplify the expression.
$\frac{x-9}{9-x}$
Select the correct choice below and fill in any answer boxes in your choice.
A. $\frac{x-9}{9-x}=$ $\square$
B. The expression cannot be simplified.
Answer (2)
Factor out -1 from the denominator: 9 − x = − ( x − 9 ) .
Rewrite the expression: 9 − x x − 9 = − ( x − 9 ) x − 9 .
Cancel the common factor ( x − 9 ) : − ( x − 9 ) x − 9 = − 1 .
The simplified expression is − 1 .
Explanation
Understanding the Problem We are asked to simplify the expression 9 − x x − 9 .
Rewriting the Denominator Notice that the denominator 9 − x is the negative of the numerator x − 9 . We can rewrite the denominator by factoring out a − 1 : 9 − x = − ( x − 9 ) .
Substituting the Rewritten Denominator Now we can rewrite the original expression as: 9 − x x − 9 = − ( x − 9 ) x − 9
Canceling the Common Factor We can cancel the common factor of ( x − 9 ) from the numerator and the denominator, provided that x = 9 : − ( x − 9 ) x − 9 = − 1 1 = − 1
Final Answer Therefore, the simplified expression is − 1 .
Examples
Simplifying rational expressions is a fundamental skill in algebra, with applications in various fields. For instance, in physics, you might encounter similar expressions when dealing with rates of change or ratios of physical quantities. Imagine you're calculating the efficiency of a machine, where efficiency is expressed as the ratio of output energy to input energy. If the expression for efficiency involves terms like 5 − x x − 5 , simplifying it to -1 (assuming x = 5 ) makes the calculation straightforward, indicating a complete energy loss or a flawed model.
The expression 9 − x x − 9 simplifies to − 1 for any value of x except 9. This happens because the denominator can be rewritten as the negative of the numerator. By canceling the common factor, we arrive at the final simplified result of − 1 .
;
