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?

A logistics software formula calculates the ratio of (Actual Weight - Target Weight) divided by (Target Weight - Actual Weight). If the Actual Weight is not equal to the Target Weight, this ratio will always simplify to the constant value of ____.

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 $-1$.