Problem 79 - Beginning and Intermediate Algebra

Question

Determine whether each statement is true or false. \(|-6|<|-9|\)

Step-by-Step Solution

Answer

True. \(|-6| < |-9|\) because 6 is less than 9.

Step 1: Understand Absolute Value

Absolute value refers to the distance of a number from zero on the number line. It does not take the sign into account, meaning \(|-x| = x\) for any number x.

Step 2: Calculate Absolute Values

Find the absolute value of each number in the inequality. For \(|-6|\), it becomes 6, and for \(|-9|\), it becomes 9.

Step 3: Compare Absolute Values

Compare the absolute values obtained: 6 and 9. Since 6 is less than 9, the original inequality \(|-6| < |-9|\) holds true.

Step 4: Final Answer

State whether the original inequality is true based on the comparison. Therefore, \(|-6| < |-9|\) is true.

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics that measures the distance of a number from zero on the number line. This measurement does not consider whether the number is positive or negative. Instead, it purely represents the magnitude of the number.

For example, the absolute value of -6 is \(|-6| = 6\), and the absolute value of 6 is \(|6| = 6\). Both have the same absolute value because they are both 6 units away from zero on the number line.

Understanding absolute value is crucial, especially when solving problems involving inequalities.

Using the Number Line

A number line is a visual tool that helps us see the relationship between numbers, including their absolute values. By representing negative and positive numbers in a linear fashion, we can easily measure distances between them and zero.

To better understand the exercise, consider placing -6 and -9 on the number line. Both values are to the left of zero, representing their negative nature, but their absolute values reveal their distances from zero.

On the number line:

• -6 is six units from zero
• -9 is nine units from zero

Thus, it becomes clear that -6 is closer to zero than -9, which means the absolute value of -6 is less than the absolute value of -9.

Comparing Inequalities

When comparing absolute values, we are effectively comparing their distances from zero without considering their sign. This is central to solving the inequality \(|-6| < |-9|\).

In the exercise, after calculating the absolute values:

• \[\begin{equation}|-6| = 6\end{equation}\]\
• \[\begin{equation}|-9| = 9\end{equation}\]\

We then compare these values: 6 and 9. Since 6 is obviously less than 9, the inequality remains true. This reflects in our final answer: \(|-6| < |-9|\) is a true statement.

Thus, understanding absolute value and using a number line are crucial in solving such comparisons in mathematical problems.