Problem 50 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression.\(-5|-5|\)

Step-by-Step Solution

Answer

The given expression \(-5|-5|\) evaluates to -25.

Step 1: Find Absolute Value

The first step is to evaluate the absolute value part of the expression which is |-5|. The absolute value of a number n, denoted by |n|, is the non-negative value of n without regard to its sign. So, |-5| is equal to 5 because the distance of -5 from zero on the number line is 5 units.

Step 2: Multiplication

The next step, we carry out the multiplication of -5 with the absolute value of -5 (which is 5). That is to multiply -5 \(\times\) 5.

Step 3: Result

The multiplication of -5 and 5 gives us -25. Hence, the expression \( -5|-5| \) evaluates to -25.

Key Concepts

Understanding the Number Line

The number line is a visual representation of numbers on a straight, horizontal line. Each point on the line corresponds to a real number.
When working with absolute values, the number line helps us understand the distance of numbers from zero. It doesn't matter if the number is positive or negative; we only consider how far it is from zero.
In the given exercise, |-5|, the number -5 lies to the left of zero on the number line. We measure its distance from zero, which is 5 units. Thus, the absolute value of -5 is 5, as it has covered 5 steps either to the left or right from zero. The number line helps make the abstract concept of absolute value more concrete.

Simplifying Multiplication

Multiplication is a key mathematical operation that involves combining groups of equal size. In simpler terms, it's like repeated addition.
For example, if we multiply -5 by 5, we are essentially adding -5 together five times:

• -5 + -5 + -5 + -5 + -5

This repeated addition gives us the product of -25.
This exercise involves multiplying a negative number with a positive one. A negative times a positive will always result in a negative number, following the rule of signs in multiplication. This rule states:

• Positive x Positive = Positive
• Negative x Negative = Positive
• Positive x Negative = Negative
• Negative x Positive = Negative

In our case, the multiplication of -5 and 5 yields -25 due to the rule of signs.

Evaluating Expressions

Expression evaluation is the process of simplifying or calculating the value of a mathematical expression. It involves following a series of steps or operations defined by math rules. The order in which these operations are performed is crucial.
For the given expression \(-5|-5|\), we evaluate it by first computing the absolute value of -5, resulting in 5. This step is essential because it transforms part of the expression into a multiplication problem.
Next, we multiply the result with -5, adhering to the rules of mathematics we learned from the multiplication section. It’s important to follow the procedures step by step:

• Compute any absolute values or parentheses first
• Perform multiplications or divisions
• Followed by any additions or subtractions

This methodology helps maintain accuracy in resolving expressions and confirms the final result of \(-25\) in our original problem.