Evaluate \[$|5-15|\$\].A. 10 B. -20 C. 20 D. -10
Understanding Absolute Value
Absolute value is a mathematical concept that represents the distance of a number from zero on the number line. It is denoted by two vertical lines, |x|, where x is the value inside the absolute value bars. The absolute value of a number is always non-negative, regardless of whether the number itself is positive or negative.
Evaluating Absolute Value Expressions
When evaluating absolute value expressions, we need to consider two cases: when the value inside the absolute value bars is positive, and when it is negative.
Case 1: Positive Value Inside Absolute Value Bars
If the value inside the absolute value bars is positive, then the absolute value of that value is simply the value itself. For example:
- |5| = 5
- |10| = 10
- |20| = 20
Case 2: Negative Value Inside Absolute Value Bars
If the value inside the absolute value bars is negative, then the absolute value of that value is the positive version of that value. For example:
- |-5| = 5
- |-10| = 10
- |-20| = 20
Evaluating the Expression |5-15|
Now, let's evaluate the expression |5-15|. To do this, we need to follow the order of operations (PEMDAS):
- Subtract 15 from 5: 5 - 15 = -10
- Take the absolute value of -10: |-10| = 10
Therefore, the value of the expression |5-15| is 10.
Answer Options
Based on our evaluation, we can now compare the answer options:
A. 10 B. -20 C. 20 D. -10
The correct answer is A. 10.
Conclusion
Frequently Asked Questions
Q: What is the absolute value of a number?
A: The absolute value of a number is the distance of that number from zero on the number line. It is always non-negative, regardless of whether the number itself is positive or negative.
Q: How do I evaluate an absolute value expression?
A: To evaluate an absolute value expression, you need to consider two cases: when the value inside the absolute value bars is positive, and when it is negative. If the value is positive, the absolute value is simply the value itself. If the value is negative, the absolute value is the positive version of that value.
Q: What is the order of operations for evaluating absolute value expressions?
A: The order of operations for evaluating absolute value expressions is:
- Evaluate the expression inside the absolute value bars.
- Take the absolute value of the result.
Q: How do I handle negative numbers inside absolute value bars?
A: When a negative number is inside absolute value bars, you need to take the positive version of that number. For example, |-5| = 5.
Q: Can I simplify absolute value expressions?
A: Yes, you can simplify absolute value expressions by evaluating the expression inside the absolute value bars and then taking the absolute value of the result.
Q: What is the difference between absolute value and distance?
A: Absolute value and distance are related concepts, but they are not exactly the same thing. Distance refers to the actual length between two points, while absolute value refers to the distance of a number from zero on the number line.
Q: Can I use absolute value to solve equations?
A: Yes, you can use absolute value to solve equations. For example, |x| = 5 can be solved by finding the values of x that are 5 units away from zero on the number line.
Q: What are some common applications of absolute value?
A: Absolute value has many applications in mathematics, science, and engineering. Some common applications include:
- Modeling real-world phenomena, such as temperature and distance
- Solving equations and inequalities
- Analyzing data and making predictions
- Optimizing functions and solving optimization problems
Q: Can I use absolute value to solve inequalities?
A: Yes, you can use absolute value to solve inequalities. For example, |x| < 5 can be solved by finding the values of x that are less than 5 units away from zero on the number line.
Q: What is the relationship between absolute value and square root?
A: The absolute value and square root functions are related, but they are not exactly the same thing. The absolute value function takes the distance of a number from zero, while the square root function takes the square root of a number.
Q: Can I use absolute value to solve systems of equations?
A: Yes, you can use absolute value to solve systems of equations. For example, |x| + |y| = 5 can be solved by finding the values of x and y that satisfy the equation.
Q: What are some common mistakes to avoid when working with absolute value?
A: Some common mistakes to avoid when working with absolute value include:
- Forgetting to take the absolute value of a negative number
- Not following the order of operations
- Not considering both cases (positive and negative) when evaluating absolute value expressions
Conclusion
In this article, we answered some frequently asked questions about evaluating absolute value expressions. We covered topics such as the definition of absolute value, the order of operations, and common applications of absolute value. We also discussed some common mistakes to avoid when working with absolute value.