Simplifying $\frac{x-8}{8-x}$

Simplifying $\frac{14-2x}{x^2-49}$

In a quality control formula, a technician encounters the ratio (Measured - Target) / (Target - Measured). What constant value does this ratio simplify to, assuming the Measured value does not equal the Target value?

Financial Ratio Simplification

In financial reporting, formulas often involve calculating the difference between two values. Based on the rules for opposites in rational expressions, match each algebraic term from a spreadsheet audit with its simplified constant or equivalent expression.

An operations analyst is auditing a spreadsheet formula for budget variance: (Budgeted Amount - Actual Spent) / (Actual Spent - Budgeted Amount). Arrange the following steps in the correct logical order to demonstrate how this rational expression simplifies to -1.

Energy Efficiency Audit

Simplifying Opposites in Business Formulas

An automated system for a manufacturing plant calculates a 'Difference Ratio' between two distinct pressure readings, P1 and P2, using the formula: (P1 - P2) / (P2 - P1). According to the rules for opposites in rational expressions, what constant value will this ratio always simplify to, assuming P1 and P2 are not equal?

An inventory manager is auditing a spreadsheet formula used to calculate a variance ratio: $\frac{\text{Projected} - \text{Actual}}{\text{Actual} - \text{Projected}}$. To explain why this ratio simplifies to $-1$, which algebraic identity correctly identifies the relationship between the numerator and the denominator?

Example of Simplifying a Rational Expression with Opposite Factors

When the numerator and denominator of a rational expression are opposites of each other, what does the expression simplify to?

Simplifying $\frac{x^2-4x-32}{64-x^2}$

If $a \ne b$, then $\frac{a-b}{b-a}$ simplifies to $-1$.