Problem 83 - Beginning and Intermediate Algebra

Question

Determine whether each statement is true or false. \(-|-5| \geq-|-9|\)

Step-by-Step Solution

Answer

True

Step 1: Understand Absolute Value

Recall that the absolute value of a number is its distance from zero on the number line, regardless of direction. For any number x, \(-|x| = -|x|\).

Step 2: Calculate Absolute Values

Determine the absolute values of -5 and -9: \(|-5| = 5\) and \(|-9| = 9\). Therefore, \(-|-5| = -5\) and \(-|-9| = -9\).

Step 3: Compare the Values

Now compare the values: Is \(-5 \geq -9\) true or false? Since -5 is greater than -9 on the number line, this statement is true.

Key Concepts

absolute value

The absolute value of a number refers to its distance from zero on the number line, without considering the direction.
It is always a non-negative number. For example, the absolute value of both 5 and -5 is 5, denoted as \(|5| = 5\). This concept of distance helps in simplifying expressions and solving equations.
When dealing with absolute values, the symbol |x| is used to signify the absolute magnitude of the number x.

number line

A number line is a straight, horizontal line that illustrates numbers at equal intervals.
It extends infinitely in both positive and negative directions. Zero is positioned at the center, serving as a reference point.
Positive numbers are to the right of zero, while negative numbers are to the left:

• Each point on the number line corresponds to a unique real number.
• The distance between any two points on the line represents the difference in their values.
• Absolute values can be easily compared using a number line.

For example, on the number line, -5 is to the left of -9, indicating that -5 is greater than -9.

inequalities

Inequalities are mathematical expressions that show the relationship between two values where they are not equal. Common symbols include:

• \(>\) meaning greater than
• \(<\) meaning less than
• \(\geq\) meaning greater than or equal to
• \(\leq\) meaning less than or equal to

Inequalities are used to compare numbers or expressions and are displayed on a number line.
For example, in the statement \(-5 \geq -9\), it means -5 is greater than or equal to -9. This is true because, on the number line, -5 is positioned to the right of -9.