Simplifying Rational Expressions and Integer Arithmetic

Problem Statement and Overview

This study guide provides a detailed, step-by-step arithmetic evaluation of a rational expression with signed integers, grouping symbols, and fraction bars. The expression to simplify and evaluate is: $\frac{56-68}{10-6} - \frac{4(3-10)}{11-13}$

Step-by-Step Evaluation of the First Fraction

To evaluate the first term $\frac{56-68}{10-6}$, simplify the numerator and denominator independently before performing the final quotient.

First, compute the subtraction in the numerator: $56 - 68 = -12$

Second, compute the subtraction in the denominator: $10 - 6 = 4$

Third, divide the simplified numerator by the simplified denominator using the rules of integer division: $\frac{-12}{4} = -3$

Step-by-Step Evaluation of the Second Fraction

To evaluate the second term $\frac{4(3-10)}{11-13}$, apply the order of operations by resolving inner parentheses first, followed by multiplication, and then evaluating the denominator.

First, perform the operation inside the parentheses in the numerator: $3 - 10 = -7$

Second, multiply the result by the outer coefficient of $4$: $4 \times (-7) = -28$

Third, evaluate the denominator: $11 - 13 = -2$

Fourth, divide the complete numerator by the denominator. Dividing a negative integer by a negative integer yields a positive quotient: $\frac{-28}{-2} = 14$

Final Combination and Solution

Substitute the simplified values of both fractional terms back into the main subtraction expression: $\left(\frac{56-68}{10-6}\right) - \left(\frac{4(3-10)}{11-13}\right) = (-3) - (14)$

Subtracting a positive value $14$ from a negative value $-3$ combines their absolute values negatively: $-3 - 14 = -17$

Thus, the final simplified value of the entire expression is $-17$.

Mathematical Principles and Rules Applied

Fraction bars function as implicit grouping symbols surrounding both the entire numerator and the entire denominator. All operations above and below the horizontal fraction line must be resolved completely before dividing.

When subtracting integers where the subtracted magnitude exceeds the starting value, the result is negative, such as $56 - 68 = -12$, $3 - 10 = -7$, and $11 - 13 = -2$.

For multiplication and division of signed real numbers, terms with opposite signs produce a negative result (e.g., $4 \times (-7) = -28$ and $\frac{-12}{4} = -3$), whereas terms with identical signs produce a positive result (e.g., $\frac{-28}{-2} = 14$).