What Is The Solution To $4|0.5x - 2.5| = 0$?A. $x = 1.25$ B. $x = 5$ C. $x = -1.25$ Or $x = 1.25$ D. $x = -5$ Or $x = 5$
Introduction
In mathematics, absolute value equations are a type of equation that involve absolute values. These equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving absolute value equations, specifically the equation $4|0.5x - 2.5| = 0.$ We will break down the solution step by step and provide a clear explanation of each step.
Understanding Absolute Value Equations
Before we dive into solving the equation, let's first understand what absolute value equations are. An absolute value equation is an equation that involves the absolute value of an expression. The absolute value of an expression is its distance from zero on the number line, without considering direction. In other words, it is the value of the expression without considering whether it is positive or negative.
The Equation $4|0.5x - 2.5| = 0$
The equation we are given is $4|0.5x - 2.5| = 0.$ To solve this equation, we need to isolate the absolute value expression. We can start by dividing both sides of the equation by 4, which gives us $|0.5x - 2.5| = 0.$
Solving the Absolute Value Equation
Now that we have isolated the absolute value expression, we can solve the equation. To do this, we need to consider two cases: when the expression inside the absolute value is positive, and when it is negative.
Case 1: When the Expression Inside the Absolute Value is Positive
When the expression inside the absolute value is positive, we can remove the absolute value sign and solve the equation. In this case, we have $0.5x - 2.5 = 0.$ To solve for x, we can add 2.5 to both sides of the equation, which gives us $0.5x = 2.5.$ Then, we can divide both sides of the equation by 0.5, which gives us $x = 5.$
Case 2: When the Expression Inside the Absolute Value is Negative
When the expression inside the absolute value is negative, we can remove the absolute value sign and solve the equation. In this case, we have $0.5x - 2.5 = 0.$ To solve for x, we can add 2.5 to both sides of the equation, which gives us $0.5x = 2.5.$ Then, we can divide both sides of the equation by 0.5, which gives us $x = 5.$
Conclusion
In conclusion, the solution to the equation $4|0.5x - 2.5| = 0$ is $x = 5.$ This is because when the expression inside the absolute value is positive, we can remove the absolute value sign and solve the equation, which gives us $x = 5.$ Similarly, when the expression inside the absolute value is negative, we can remove the absolute value sign and solve the equation, which also gives us $x = 5.$
Comparison with Answer Choices
Now that we have solved the equation, let's compare our solution with the answer choices. The answer choices are A. , B. , C. or , and D. or . Our solution, $x = 5$, matches answer choice B.
Final Answer
Introduction
In our previous article, we discussed how to solve absolute value equations, specifically the equation $4|0.5x - 2.5| = 0.$ We broke down the solution step by step and provided a clear explanation of each step. In this article, we will answer some frequently asked questions related to solving absolute value equations.
Q: What is an absolute value equation?
A: An absolute value equation is an equation that involves the absolute value of an expression. The absolute value of an expression 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. Then, you can consider two cases: when the expression inside the absolute value is positive, and when it is negative. For each case, you can remove the absolute value sign and solve the equation.
Q: What is the difference between an absolute value equation and a linear equation?
A: An absolute value equation is an equation that involves the absolute value of an expression, while a linear equation is an equation that involves a linear expression. For example, the equation $|x - 2| = 3$ is an absolute value equation, while the equation $x - 2 = 3$ is a linear equation.
Q: Can I use the same method to solve all absolute value equations?
A: No, you cannot use the same method to solve all absolute value equations. The method we discussed in our previous article is for equations of the form $|ax - b| = c.$ If the equation has a different form, you may need to use a different method to solve it.
Q: What if the equation has a variable inside the absolute value?
A: If the equation has a variable inside the absolute value, you can use the same method we discussed in our previous article. However, you may need to use algebraic manipulations to isolate the variable.
Q: Can I use a calculator to solve absolute value equations?
A: Yes, you can use a calculator to solve absolute value equations. However, it's always a good idea to check your work by hand to make sure you get the correct solution.
Q: What if I get multiple solutions to an absolute value equation?
A: If you get multiple solutions to an absolute value equation, it means that the equation has multiple solutions. This can happen when the equation has a variable inside the absolute value, or when the equation has a coefficient of 1.
Q: How do I know which solution to choose?
A: To choose the correct solution, you need to check each solution in the original equation. If a solution satisfies the original equation, it is a valid solution. If a solution does not satisfy the original equation, it is not a valid solution.
Conclusion
In conclusion, solving absolute value equations can be challenging, but with the right approach, it can be done with ease. We hope this article has helped you understand how to solve absolute value equations and has answered some of your frequently asked questions.
Additional Resources
If you want to learn more about solving absolute value equations, we recommend checking out the following resources:
- Khan Academy: Absolute Value Equations
- Mathway: Absolute Value Equations
- Wolfram Alpha: Absolute Value Equations
Final Answer
The final answer is B. .