Problem 85 - Beginning and Intermediate Algebra

Question

Determine whether each statement is true or false. \(|6-5| \geq|6-2|\)

Step-by-Step Solution

Answer

False

Step 1 - Evaluate the Absolute Values

Calculate the absolute values of the expressions inside the vertical bars. For \(|6-5|\), this simplifies to \(|1|\), which is 1. For \(|6-2|\), this simplifies to \(|4|\), which is 4.

Step 2 - Compare the Absolute Values

Compare the results from Step 1. Determine if 1 is greater than or equal to 4.

Step 3 - Conclusion

Since 1 is not greater than or equal to 4, the statement \(|6-5| \geq |6-2|\) is false.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero, regardless of direction. Essentially, it tells us how far a number is from zero on a number line.
For instance, the absolute value of both -3 and 3 is 3 because they are both 3 units away from zero.
Absolute values are commonly represented using vertical bars, like \(|x|\). Here, for example, \(|6-5|\) simplifies to \(|1|\), which equals 1.
Likewise, \(|6-2|\) simplifies to \(|4|\), which equals 4.
To recap:

• \(|x|\) simplifies to \(x\) when \(x\) is positive or zero.
• \(|x|\) simplifies to \(-x\) when \(x\) is negative.

Inequality

An inequality compares two values, showing if one is less than, greater than, less than or equal to, or greater than or equal to another. Symbols like \(<\), \(>\), \(\leq\), and \(\geq\) are used for inequalities.
In the given exercise, the inequality \(\geq\) means 'greater than or equal to.' It's essential to evaluate each side of the inequality separately before comparing them.
In our example, we have the statement \(|6-5| \geq |6-2|\). By calculating the absolute values, we get 1 and 4 respectively. Now, we compare these values to determine the truth of the statement.
Since 1 is not greater than or equal to 4, the statement is false. This demonstrates that carefully evaluating each part of an inequality is crucial to finding the correct answer.

Evaluate Expressions

Evaluating expressions is about simplifying them to their most basic form. This involves performing operations like addition, subtraction, and finding absolute values.
In the given problem, we start with expressions inside the absolute value signs: \(6-5\) and \(6-2\). Simplifying these gives us 1 and 4.
Next, we find the absolute values: \(|1|\) and \(|4|\), which are 1 and 4 respectively.
Finally, we compare these values using the inequality sign to determine if the original statement holds true.
This process involves:

• Breaking down complex expressions into simpler parts.
• Applying absolute value rules to simplify.
• Using the simplified results to evaluate the inequality or expression.

By following these steps, we ensure accuracy in our mathematical evaluations.