Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression.\(\frac{-5}{|-5|}\)

Step-by-Step Solution

Answer

-1

Step 1: Resolve the Absolute Value

Compute the absolute value |-5|. The absolute value of a number is its distance from zero. In this case, |-5| equals to 5.

Step 2: Simplify the Expression

Now that we have the value of the absolute in the denominator as 5, we substitute it into the original expression. Thus, the fraction becomes \(-5/5\).

Step 3: Compute the Fraction

We carry out the division operation. -5 divided by 5 is -1.

Key Concepts

Algebra

Algebra is the branch of mathematics that uses symbols, usually letters, to represent numbers in equations and formulas. These symbols can stand for unknown values, or variables, that we seek to solve for. An understanding of basic algebra is essential in mathematics and plays a critical role in solving more complex problems.

In our example, we come across an algebraic expression involving an absolute value. Algebra is all about finding the definitive answer to the expression by applying the correct operations, which in this case include dealing with an absolute value and simplifying a fraction. By breaking down algebraic expressions into simpler components, we can solve them step by step, as seen in the provided exercise.

Absolute Value Operations

The absolute value of a number is a measure of its magnitude or distance from zero on the number line, regardless of its direction. In mathematical notation, the absolute value of a number 'a' is denoted as \( |a| \). It is always a non-negative value.

Understanding how to perform operations with absolute values is critical in many areas of algebra. When you're given an expression like \( \frac{-5}{|-5|} \), the first step is to evaluate the absolute value in the denominator. Once the absolute value of -5 is recognized as 5, we can replace \( |-5| \), in the expression with 5. This process demonstrates an important element of working with absolute values: turning negative numbers into their positive counterparts to further evaluate or simplify an expression.

Simplifying Expressions

Simplifying expressions is a fundamental technique in algebra that makes equations easier to work with. By breaking an expression down into its simplest form, we not only make it easier to understand but often make the path to the solution clearer.

In the given exercise, after dealing with the absolute value, we simplify the fraction \( -5/5 \). Since the numerator and denominator are equal in magnitude but opposite in sign, we simplify to -1. This step is crucial because it takes us to the final answer and exemplifies a key skill in algebra: reducing expressions to their simplest terms. Simplification often involves combining like terms, reducing fractions, and cancelling out factors that appear in both the numerator and the denominator, just to name a few techniques.