Drag The Tiles To The Correct Boxes.Put The Equations In Order From Least To Greatest Number Of Solutions:1. { |x-8|-4=-1$}$2. { \frac{1}{2}|x|+3=3$}$3. ${ 3-|x+4|=10\$}
Introduction
Absolute value equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of absolute value. In this article, we will explore three absolute value equations and solve them step by step. We will also discuss the concept of the number of solutions and how to determine the order of the equations from least to greatest number of solutions.
What are Absolute Value Equations?
Absolute value equations are equations that involve the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction. For example, the absolute value of -3 is 3, and the absolute value of 3 is also 3.
Equation 1: |x-8|-4=-1
To solve this equation, we need to isolate the absolute value expression. We can do this by adding 4 to both sides of the equation:
|x-8|-4=-1 |x-8|=-1+4 |x-8|=3
Now, we need to consider two cases: when x-8 is positive and when x-8 is negative.
Case 1: x-8 is positive
When x-8 is positive, we can remove the absolute value sign and solve for x:
x-8=3 x=3+8 x=11
Case 2: x-8 is negative
When x-8 is negative, we can remove the absolute value sign and solve for x:
-(x-8)=3 -x+8=3 -x=3-8 -x=-5 x=5
Therefore, the solutions to the equation |x-8|-4=-1 are x=11 and x=5.
Equation 2: 1/2|x|+3=3
To solve this equation, we need to isolate the absolute value expression. We can do this by subtracting 3 from both sides of the equation:
1/2|x|+3=3 1/2|x|=3-3 1/2|x|=0
Now, we need to consider two cases: when x is positive and when x is negative.
Case 1: x is positive
When x is positive, we can remove the absolute value sign and solve for x:
1/2x=0 x=0
Case 2: x is negative
When x is negative, we can remove the absolute value sign and solve for x:
-(1/2x)=0 -1/2x=0 x=0
However, we need to consider the case when x is zero. When x is zero, the absolute value expression is also zero. Therefore, the solution to the equation 1/2|x|+3=3 is x=0.
Equation 3: 3-|x+4|=10
To solve this equation, we need to isolate the absolute value expression. We can do this by subtracting 3 from both sides of the equation:
3-|x+4|=10 -|x+4|=10-3 -|x+4|=7
Now, we need to consider two cases: when x+4 is positive and when x+4 is negative.
Case 1: x+4 is positive
When x+4 is positive, we can remove the absolute value sign and solve for x:
-x-4=7 -x=7+4 -x=11 x=-11
Case 2: x+4 is negative
When x+4 is negative, we can remove the absolute value sign and solve for x:
-(x+4)=7 -x-4=7 -x=7+4 -x=11 x=-11
However, we need to consider the case when x+4 is zero. When x+4 is zero, the absolute value expression is also zero. Therefore, the solution to the equation 3-|x+4|=10 is x=-11.
Ordering the Equations from Least to Greatest Number of Solutions
Now that we have solved all three equations, we can determine the order of the equations from least to greatest number of solutions.
Equation 1 has two solutions: x=11 and x=5.
Equation 2 has one solution: x=0.
Equation 3 has one solution: x=-11.
Therefore, the order of the equations from least to greatest number of solutions is:
- Equation 2: 1/2|x|+3=3 (one solution)
- Equation 3: 3-|x+4|=10 (one solution)
- Equation 1: |x-8|-4=-1 (two solutions)
Conclusion
Q: What is an absolute value equation?
A: An absolute value equation is an equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction.
Q: How do I solve an absolute value equation?
A: To solve an absolute value equation, you need to isolate the absolute value expression and then consider two cases: when the expression inside the absolute value is positive and when it is negative. You can then remove the absolute value sign and solve for the variable.
Q: What are the two cases I need to consider when solving an absolute value equation?
A: The two cases you need to consider are:
- When the expression inside the absolute value is positive.
- When the expression inside the absolute value is negative.
Q: How do I determine which case to consider first?
A: To determine which case to consider first, you need to look at the expression inside the absolute value and determine whether it is positive or negative. If it is positive, you can consider the first case. If it is negative, you can consider the second case.
Q: What if the expression inside the absolute value is zero?
A: If the expression inside the absolute value is zero, you need to consider both cases. In this case, the absolute value expression is equal to zero, and you can remove the absolute value sign and solve for the variable.
Q: How do I determine the number of solutions to an absolute value equation?
A: To determine the number of solutions to an absolute value equation, you need to consider the two cases and count the number of solutions in each case. If there are solutions in both cases, you need to count them separately.
Q: Can an absolute value equation have more than two solutions?
A: Yes, an absolute value equation can have more than two solutions. This can happen when the expression inside the absolute value is a quadratic expression or when there are multiple solutions in one of the cases.
Q: How do I order absolute value equations from least to greatest number of solutions?
A: To order absolute value equations from least to greatest number of solutions, you need to count the number of solutions to each equation and then compare them. The equation with the fewest number of solutions comes first, and the equation with the most number of solutions comes last.
Q: What are some common mistakes to avoid when solving absolute value equations?
A: Some common mistakes to avoid when solving absolute value equations include:
- Not isolating the absolute value expression
- Not considering both cases
- Not counting the number of solutions correctly
- Not ordering the equations correctly
Q: How can I practice solving absolute value equations?
A: You can practice solving absolute value equations by working through examples and exercises in a textbook or online resource. You can also try solving absolute value equations on your own and then checking your answers with a calculator or online tool.
Q: What are some real-world applications of absolute value equations?
A: Absolute value equations have many real-world applications, including:
- Physics: to model the motion of objects
- Engineering: to design and optimize systems
- Economics: to model economic systems and make predictions
- Computer Science: to solve problems in computer graphics and game development
By following these tips and practicing regularly, you can become proficient in solving absolute value equations and apply them to real-world problems.