A student in an algebra review course is simplifying the following expression:

$\frac{6x^2-x+20}{x^2-81} - \frac{5x^2+11x-7}{x^2-81}$

Arrange the steps below in the correct chronological order to complete the subtraction and simplify the result.

As part of a continuous improvement training program, you are reviewing algebraic models used to calculate the difference in projected regional costs. The difference is modeled by the following expression: $\frac{6x^2-x+20}{x^2-81} - \frac{5x^2+11x-7}{x^2-81}$. When performing the first step of subtracting the rational expressions, you must recall the procedural rule to distribute the subtraction operation across the entire second numerator. Which of the following correctly displays the combined numerator immediately after distributing the negative sign, but before combining any like terms?

A logistics manager is simplifying a formula that represents the difference in delivery times between two regional routes: $\frac{6x^2-x+20}{x^2-81} - \frac{5x^2+11x-7}{x^2-81}$. True or False: When this expression is simplified to its lowest terms, the denominator of the resulting rational expression is $x+9$.

A logistics coordinator is comparing two cost-efficiency models represented by the rational expressions $\frac{6x^2-x+20}{x^2-81}$ and $\frac{5x^2+11x-7}{x^2-81}$. To find the difference between these models, the coordinator must subtract the two expressions and simplify. Match each algebraic component of the simplification process to its corresponding form.

Intermediate Simplification of Quality Control Models

An operations analyst at an energy company is comparing two efficiency models. The difference in performance between the two systems over a variable workload $x$ is given by the expression:

$\frac{6x^2-x+20}{x^2-81} - \frac{5x^2+11x-7}{x^2-81}$

To find the difference, the analyst begins by writing the subtraction of the numerators over the common denominator:

$\frac{6x^2-x+20 - (5x^2+11x-7)}{x^2-81}$

To simplify the expression, the analyst must recall the rule for distributing the subtraction sign across the second numerator. When the negative sign is distributed to each term of the polynomial ${}5x^2+11x-7$ inside the parentheses, the middle term $+11x$ becomes ____. (Enter your answer as a term, such as -11x, with no spaces)

Auditing a Cost-Difference Formula