Problem 80 - Beginning and Intermediate Algebra

Question

Determine whether each statement is true or false. \(|-12|<|-20|\)

Step-by-Step Solution

Answer

True

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, \(|-x| = x\) for any positive number x.

Step 2: Calculate Absolute Values

Find the absolute values of -12 and -20. \(|-12| = 12\) and \(|-20| = 20\).

Step 3: Compare the Absolute Values

Compare the absolute values obtained in the previous step. Since 12 is less than 20, \(|-12| < |-20|\).

Step 4: Determine the Truth Value

Since the statement \(|-12| < |-20|\) is correct based on the absolute values calculated, the statement is true.

Key Concepts

absolute value

Absolute value is one of the fundamental concepts in mathematics. It refers to the distance of a number from zero on the number line, without considering the direction. This means that both positive and negative numbers have the same absolute value if they are the same distance from zero. For example, \( -12 \) and 12 both have an absolute value of 12. You can think of absolute value as a way to measure how far a number is from zero.
When dealing with absolute values, here's what to remember:

• The absolute value of a positive number is the number itself, such as \(|5| = 5\)
• The absolute value of a negative number is its positive counterpart, for example, \(|-3| = 3\)
• The absolute value of zero is zero, meaning \(|0| = 0\)

This is why we see in the problem that the absolute value of \(|-12| = 12\) and \(|-20| = 20\). Understanding this is crucial for comparing the sizes of numbers based on their absolute values.

number line

The number line is a visual representation of numbers placed in order of their value. Numbers to the right are larger (more positive), and numbers to the left are smaller (more negative).
Each point on the number line corresponds to a unique number. Absolute values help us understand numbers in terms of their distance from zero.
To visualize this, consider the number line:
... -3, -2, -1, 0, 1, 2, 3 ...
On this line, the number -3 is three steps to the left of zero, so its absolute value is 3. The number 3 is three steps to the right of zero, so its absolute value is also 3. They are the same distance from zero, just in opposite directions. When comparing absolute values, you are essentially looking at these distances without worrying about direction.
In our example, -12 and -20 both lie to the left of zero. But since \(|-12| = 12\) and \(|-20| = 20\), -20 is farther from zero than -12. So, on the number line, 12 (the absolute value of -12) is indeed less than 20 (the absolute value of -20). This makes the statement \(|-12| < |-20|\) true.

inequality

Inequality is used to compare two values. It helps us understand the relationship between numbers and whether one is less than, greater than, or equal to another. Specifically, inequalities can be expressed using symbols:

• < (less than)
• > (greater than)
• \(eq\) (not equal to)
• \( \leq\) (less than or equal to)
• \( \geq\) (greater than or equal to)

When you compare absolute values, you are determining whether one is larger or smaller than another without regard to their original signs. In the case of \(|-12| < |-20|\), you are saying that the distance from zero of \(-12\) is less than the distance from zero of \(-20\).
This helps to solve the inequality by comparing absolute terms: \(12 < 20\) is true, and therefore, \(|-12| < |-20|\) is true. Inequalities like this are useful in many areas of mathematics and applied sciences because they help us quantify and compare if one value is larger or smaller than another.