Problem 45 - Calculus with Concepts in Calculus

Question

Evaluate the expression. $$ |-5|+|5| $$

Step-by-Step Solution

Answer

The expression evaluates to 10.

Step 1: Understand the Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Thus, the absolute value is always non-negative. We need to find the absolute value of each term in the expression separately.

Step 2: Calculate the Absolute Value of -5

For the term \(-5\), the absolute value is obtained by ignoring the negative sign. Therefore, \(|-5| = 5\).

Step 3: Calculate the Absolute Value of 5

For the term \(5\), the number is already positive, so its absolute value does not change. Therefore, \(|5| = 5\).

Step 4: Add the Absolute Values

Now add the absolute values that were calculated earlier: \(5 + 5\). This results in \(10\).

Key Concepts

Number Line

A number line is a visual representation of numbers placed in order along a straight line. It extends infinitely in both directions, with zero typically placed at the center. Numbers increase as you move to the right and decrease as you move to the left. This tool is invaluable in helping understand concepts such as positive and negative numbers, as well as absolute value.

• The center point, or origin, of the number line is zero.
• To the right of zero, you find positive numbers which increase in magnitude.
• To the left of zero, you find negative numbers which also increase in magnitude as you move further left (e.g., -3 is further from zero than -1).
• Number lines can be helpful for visualizing operations such as addition, subtraction, and even absolute values, where you look at the distance a number is from zero without considering direction.

Understanding how to use a number line not only aids in the comprehension of absolute value but also in a wide array of mathematical operations.

Negative Numbers

Negative numbers are values found on the left side of zero on a number line. They are represented by a minus sign (-) and signify values less than zero. Negative numbers are fundamental in mathematics, representing debts, temperatures below zero, and more.

• For absolute value, negative numbers are viewed without their negative sign, as the absolute value focuses on distance from zero.
• When calculating absolute values, the negative sign simply represents direction on a number line, which is ignored.
• The larger the negative number, the further it is from zero on the number line. For instance, -5 is further from zero than -2.

By understanding negative numbers, students can better grasp concepts of subtraction, debts, and temperatures, along with comprehending how to calculate and interpret absolute values effectively.

Positive Numbers

Positive numbers are found on the right side of zero on a number line. Unlike negative numbers, they are represented without a negative sign and indicate a value that is greater than zero.

• Positive numbers do not change when considering their absolute value, as they are already counted as distances from zero in a straightforward manner.
• They are used to represent quantities, elevations above sea level, and credits in account balances, among other things.
• Since the absolute value of a positive number is the number itself, there is a more direct relationship with absolute values compared to negative numbers.

By understanding positive numbers, students can effectively engage with arithmetic operations and comprehend how these values contribute to or impact calculations like finding absolute values within expressions.