Which Statement Is True?A. The Equation $-3|2x + 1.2| = -1$ Has No Solution.B. The Equation $3.5|6x - 2| = 3.5$ Has One Solution.C. The Equation \$5|-3.1x + 6.9| = -3.5$[/tex\] Has Two Solutions.D. The Equation
Introduction
Absolute value equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, calculus, and engineering. In this article, we will delve into the world of absolute value equations and explore the different types of solutions they can have. We will examine four statements, each claiming to be true about the number of solutions to a specific absolute value equation. Our goal is to determine which statement is true and provide a comprehensive understanding of the underlying mathematics.
Understanding Absolute Value Equations
Before we dive into the statements, let's briefly review the concept of absolute value equations. An absolute value equation is an equation that contains an absolute value expression, which is denoted by the symbol | |. The absolute value of a number is its distance from zero on the number line, without considering its direction. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.
The General Form of an Absolute Value Equation
The general form of an absolute value equation is |ax + b| = c, where a, b, and c are constants. To solve an absolute value equation, we need to consider two cases:
- Case 1: ax + b = c
- Case 2: ax + b = -c
Solving Absolute Value Equations
To solve an absolute value equation, we need to isolate the absolute value expression and then consider the two cases. Let's consider the equation |2x + 1.2| = -1.
Case 1: 2x + 1.2 = -1
Solving for x, we get:
2x = -1.2 x = -0.6
Case 2: 2x + 1.2 = 1
Solving for x, we get:
2x = 0.8 x = 0.4
Analyzing the Statements
Now that we have a basic understanding of absolute value equations, let's analyze the four statements:
A. The equation $-3|2x + 1.2| = -1$ has no solution.
B. The equation $3.5|6x - 2| = 3.5$ has one solution.
C. The equation $5|-3.1x + 6.9| = -3.5$ has two solutions.
D. The equation $|2x + 1.2| = -1$ has two solutions.
Statement A: No Solution
To determine if statement A is true, we need to examine the equation $-3|2x + 1.2| = -1$. Since the absolute value of any expression is always non-negative, the left-hand side of the equation is always non-positive. However, the right-hand side of the equation is -1, which is a positive number. Therefore, the equation has no solution.
Statement B: One Solution
To determine if statement B is true, we need to examine the equation $3.5|6x - 2| = 3.5$. We can start by dividing both sides of the equation by 3.5, which gives us:
|6x - 2| = 1
Now, we can consider the two cases:
- Case 1: 6x - 2 = 1
- Case 2: 6x - 2 = -1
Solving for x in both cases, we get:
- Case 1: x = 1/6
- Case 2: x = 1/3
Since there are two distinct solutions, statement B is false.
Statement C: Two Solutions
To determine if statement C is true, we need to examine the equation $5|-3.1x + 6.9| = -3.5$. However, since the absolute value of any expression is always non-negative, the left-hand side of the equation is always non-positive. However, the right-hand side of the equation is -3.5, which is a positive number. Therefore, the equation has no solution.
Statement D: Two Solutions
To determine if statement D is true, we need to examine the equation $|2x + 1.2| = -1$. However, since the absolute value of any expression is always non-negative, the left-hand side of the equation is always non-negative. However, the right-hand side of the equation is -1, which is a negative number. Therefore, the equation has no solution.
Conclusion
In conclusion, only statement A is true. The equation $-3|2x + 1.2| = -1$ has no solution. The other three statements are false, as they claim that the equations have one or two solutions. We hope that this article has provided a comprehensive understanding of absolute value equations and their solutions.
Final Thoughts
Absolute value equations are a fundamental concept in mathematics, and they play a crucial role in various fields. In this article, we have examined four statements, each claiming to be true about the number of solutions to a specific absolute value equation. We have determined that only statement A is true, and the other three statements are false. We hope that this article has provided a valuable resource for students and professionals alike.
References
- [1] "Absolute Value Equations" by Math Open Reference
- [2] "Solving Absolute Value Equations" by Khan Academy
- [3] "Absolute Value Equations and Inequalities" by Purplemath
Glossary
- Absolute Value: The distance of a number from zero on the number line, without considering its direction.
- Absolute Value Equation: An equation that contains an absolute value expression.
- Case 1: The first case of an absolute value equation, where the expression inside the absolute value is equal to the constant.
- Case 2: The second case of an absolute value equation, where the expression inside the absolute value is equal to the negative of the constant.
- Solution: A value that satisfies an equation or inequality.
Introduction
In our previous article, we explored the concept of absolute value equations and examined four statements, each claiming to be true about the number of solutions to a specific absolute value equation. We determined that only statement A is true, and the other three statements are false. In this article, we will provide a Q&A guide to help students and professionals better understand absolute value equations.
Q: What is an absolute value equation?
A: An absolute value equation is an equation that contains an absolute value expression, which is denoted by the symbol | |. The absolute value of a number is its distance from zero on the number line, without considering its direction.
Q: How do I solve an absolute value equation?
A: To solve an absolute value equation, you need to consider two cases:
- Case 1: ax + b = c
- Case 2: ax + b = -c
You can then solve for x in both cases to find the solutions.
Q: What is the difference between Case 1 and Case 2?
A: Case 1 is when the expression inside the absolute value is equal to the constant, while Case 2 is when the expression inside the absolute value is equal to the negative of the constant.
Q: How do I determine which case to use?
A: You can determine which case to use by looking at the equation and identifying the absolute value expression. If the expression is equal to the constant, use Case 1. If the expression is equal to the negative of the constant, use Case 2.
Q: What if the equation has no solution?
A: If the equation has no solution, it means that the absolute value expression is always non-negative, and the right-hand side of the equation is a negative number. In this case, there is no value of x that can satisfy the equation.
Q: Can an absolute value equation have multiple solutions?
A: Yes, an absolute value equation can have multiple solutions. This occurs when the equation has two distinct solutions, one for each case.
Q: How do I know if an absolute value equation has one or two solutions?
A: You can determine if an absolute value equation has one or two solutions by solving the equation and checking if there are two distinct solutions.
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 considering both cases
- Not isolating the absolute value expression
- Not checking if the equation has no solution
Q: How can I practice solving absolute value equations?
A: You can practice solving absolute value equations by working through examples and exercises. You can also use online resources, such as Khan Academy or Math Open Reference, to help you practice.
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
Conclusion
In conclusion, absolute value equations are a fundamental concept in mathematics, and they have many real-world applications. By understanding how to solve absolute value equations, you can better analyze and solve problems in various fields. We hope that this Q&A guide has provided a valuable resource for students and professionals alike.
Final Thoughts
Absolute value equations are a powerful tool for solving problems in mathematics and other fields. By mastering the concept of absolute value equations, you can better analyze and solve problems, and make predictions about real-world systems. We hope that this article has provided a comprehensive guide to absolute value equations and their applications.
References
- [1] "Absolute Value Equations" by Math Open Reference
- [2] "Solving Absolute Value Equations" by Khan Academy
- [3] "Absolute Value Equations and Inequalities" by Purplemath
Glossary
- Absolute Value: The distance of a number from zero on the number line, without considering its direction.
- Absolute Value Equation: An equation that contains an absolute value expression.
- Case 1: The first case of an absolute value equation, where the expression inside the absolute value is equal to the constant.
- Case 2: The second case of an absolute value equation, where the expression inside the absolute value is equal to the negative of the constant.
- Solution: A value that satisfies an equation or inequality.