Identify the absolute value expression

Oct 13, 2025 by windailys.com 84 views

Introduction

When dealing with equations involving absolute values, it's essential to understand the different types of solutions they can have. In this article, we'll explore four equations with absolute values and determine which one has only one solution.

Understanding 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 4 is 4.

Analyzing Equation A: $|x-5|=-1$

Equation A is $|x-5|=-1$ . To solve this equation, we need to consider two cases: when $x-5$ is positive and when $x-5$ is negative.

When $x-5$ is positive, we have $x-5=-1$ , which gives us $x=4$ .
When $x-5$ is negative, we have $-(x-5)=-1$ , which gives us $x=6$ .

However, we need to check if these solutions are valid. Since $x-5$ is positive when $x>5$ , the solution $x=4$ is not valid. On the other hand, since $x-5$ is negative when $x<5$ , the solution $x=6$ is also not valid.

Analyzing Equation B: $|-6-2x|=8$

Equation B is $|-6-2x|=8$ . To solve this equation, we need to consider two cases: when $-6-2x$ is positive and when $-6-2x$ is negative.

When $-6-2x$ is positive, we have $-6-2x=8$ , which gives us $x=-7$ .
When $-6-2x$ is negative, we have $-(-6-2x)=8$ , which gives us $-6-2x=-8$ , and then $x=-1$ .

However, we need to check if these solutions are valid. Since $-6-2x$ is positive when $x<-3$ , the solution $x=-7$ is valid. On the other hand, since $-6-2x$ is negative when $x>-3$ , the solution $x=-1$ is not valid.

Analyzing Equation C: $|5x+10|=10$

Equation C is $|5x+10|=10$ . To solve this equation, we need to consider two cases: when $5x+10$ is positive and when $5x+10$ is negative.

When $5x+10$ is positive, we have $5x+10=10$ , which gives us $x=0$ .
When $5x+10$ is negative, we have $-(5x+10)=10$ , which gives us $5x+10=-20$ , and then $x=-3$ .

However, we need to check if these solutions are valid. Since $5x+10$ is positive when $x>-2$ , the solution $x=0$ is valid. On the other hand, since $5x+10$ is negative when $x<-2$ , the solution $x=-3$ is also valid.

Analyzing Equation D: $|-6x+3|=0$

Equation D is $|-6x+3|=0$ . To solve this equation, we need to consider two cases: when $-6x+3$ is positive and when $-6x+3$ is negative.

When $-6x+3$ is positive, we have $-6x+3=0$ , which gives us $x=\frac{3}{6}=\frac{1}{2}$ .
When $-6x+3$ is negative, we have $-(-6x+3)=0$ , which gives us $-6x+3=0$ , and then $x=\frac{1}{2}$ .

However, we need to check if these solutions are valid. Since $-6x+3$ is positive when $x>\frac{1}{2}$ , the solution $x=\frac{1}{2}$ is valid. On the other hand, since $-6x+3$ is negative when $x<\frac{1}{2}$ , the solution $x=\frac{1}{2}$ is also valid.

Conclusion

In conclusion, we have analyzed four equations with absolute values and determined which one has only one solution. Equation D, $|-6x+3|=0$ , has only one solution, which is $x=\frac{1}{2}$ . This is because the absolute value of $-6x+3$ is equal to zero, which means that $-6x+3$ must be equal to zero. Therefore, the only solution is $x=\frac{1}{2}$ .

Frequently Asked Questions

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 consider two cases: when the expression inside the absolute value is positive and when it is negative. You then solve each case separately and check if the solutions are valid.

Q: What is the difference between a valid and an invalid solution?

A: A valid solution is a solution that satisfies the original equation, while an invalid solution is a solution that does not satisfy the original equation.

Q: Can an absolute value equation have more than one solution?

A: Yes, an absolute value equation can have more than one solution. However, in the case of Equation D, $|-6x+3|=0$ , there is only one solution, which is $x=\frac{1}{2}$ .

Final Thoughts

In conclusion, solving absolute value equations requires careful consideration of the two cases: when the expression inside the absolute value is positive and when it is negative. By following these steps, you can determine which equation has only one solution and understand the properties of absolute value equations.

Introduction

Absolute value equations are a fundamental concept in algebra, and understanding them is crucial for solving various mathematical problems. In this article, we will delve into the world of absolute value equations, exploring their properties, solving techniques, and real-world applications.

Understanding Absolute Value Equations

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. For example, the absolute value of -3 is 3, and the absolute value of 4 is 4.

Properties of Absolute Value Equations

Absolute value equations have several properties that make them unique:

Non-Negativity: The absolute value of any number is always non-negative.
Symmetry: The absolute value of a number is the same as the absolute value of its negative counterpart.
Distance: The absolute value of a number represents its distance from zero on the number line.

Solving Absolute Value Equations

To solve an absolute value equation, you need to consider two cases:

Case 1: Expression inside the absolute value is positive: In this case, the absolute value equation becomes a simple linear equation.
Case 2: Expression inside the absolute value is negative: In this case, the absolute value equation becomes a simple linear equation with a negative sign.

Step-by-Step Guide to Solving Absolute Value Equations

Here's a step-by-step guide to solving absolute value equations:

Identify the absolute value expression: Identify the expression inside the absolute value brackets.
Consider the two cases: Consider the two cases: when the expression inside the absolute value is positive and when it is negative.
Solve each case: Solve each case separately, using the properties of absolute value equations.
Check for valid solutions: Check if the solutions obtained in each case are valid.

Real-World Applications of Absolute Value Equations

Absolute value equations have numerous real-world applications, including:

Distance and speed: Absolute value equations are used to calculate distance and speed in physics and engineering.
Finance: Absolute value equations are used to calculate the absolute value of a stock's price or a company's profit.
Computer science: Absolute value equations are used in computer science to calculate the distance between two points in a graph or a network.

Frequently Asked Questions

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.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Q: What is the difference between a valid and an invalid solution?

A: A valid solution is a solution that satisfies the original equation, while an invalid solution is a solution that does not satisfy the original equation.

Q: Can an absolute value equation have more than one solution?

A: Yes, an absolute value equation can have more than one solution.

Q: What are some real-world applications of absolute value equations?

A: Absolute value equations have numerous real-world applications, including distance and speed, finance, and computer science.

Conclusion

In conclusion, absolute value equations are a fundamental concept in algebra, and understanding them is crucial for solving various mathematical problems. By following the step-by-step guide to solving absolute value equations, you can master this concept and apply it to real-world problems.

Final Thoughts

Absolute value equations are a powerful tool for solving mathematical problems. By understanding their properties and solving techniques, you can apply them to various real-world applications. Whether you're a student, a professional, or simply someone interested in mathematics, absolute value equations are an essential concept to grasp.

Additional Resources

For further learning, here are some additional resources:

Online tutorials: Websites like Khan Academy, Mathway, and Wolfram Alpha offer interactive tutorials and exercises on absolute value equations.
Textbooks: Books like "Algebra" by Michael Artin and "Calculus" by Michael Spivak provide comprehensive coverage of absolute value equations.
Online communities: Join online forums like Reddit's r/learnmath and r/math to connect with other math enthusiasts and get help with absolute value equations.

By following these resources and practicing with exercises, you can become proficient in solving absolute value equations and apply them to real-world problems.