Not considering the sign of the expression inside the absolute value

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What are the Solutions to the Equation $|y-3|=2$ ?

The absolute value equation $|y-3|=2$ is a fundamental concept in mathematics, and solving it requires a clear understanding of absolute value and its properties. In this article, we will delve into the solutions of this equation, breaking it down into manageable steps and providing a comprehensive explanation.

Understanding Absolute Value

Before we dive into solving the equation, it's essential to understand the concept of absolute value. The absolute value of a number $x$ , denoted by $|x|$ , is the distance of $x$ from zero on the number line. In other words, it's the magnitude of $x$ without considering its direction. For example, the absolute value of $-3$ is $3$ , and the absolute value of $5$ is also $5$ .

Rewriting the Equation

To solve the equation $|y-3|=2$ , we need to rewrite it as two separate equations, one for each case: when the expression inside the absolute value is positive and when it's negative.

Case 1: $(y-3)=2$
Case 2: $-(y-3)=2$

These two cases cover all possible scenarios, and solving each one will give us the solutions to the original equation.

Solving the First Equation

Now, let's solve the first equation: $(y-3)=2$ . To isolate $y$ , we need to add $3$ to both sides of the equation.

(y-3) = 2
y - 3 + 3 = 2 + 3
y = 5

Therefore, the solution to the first equation is $y = 5$ .

Solving the Second Equation

Next, let's solve the second equation: $-(y-3)=2$ . To isolate $y$ , we need to multiply both sides of the equation by $-1$ and then add $3$ to both sides.

-(y-3) = 2
-(y-3) + (y-3) = 2 + (y-3)
-y + 3 = 2 + y - 3
-y + 3 - 3 = 2 + y - 3 - 3
-y = 2 - 3 + y
-y = -1 + y
-y - y = -1 + y - y
-2y = -1
-2y / -2 = -1 / -2
y = 1/2

Therefore, the solution to the second equation is $y = \frac{1}{2}$ .

Conclusion

In conclusion, the solutions to the equation $|y-3|=2$ are $y = 5$ and $y = \frac{1}{2}$ . These two values satisfy the original equation, and they were obtained by solving the two separate cases: when the expression inside the absolute value is positive and when it's negative.

Final Thoughts

The absolute value equation $|y-3|=2$ is a simple yet powerful concept in mathematics. By understanding the properties of absolute value and applying the correct techniques, we can solve equations like this one and gain a deeper appreciation for the beauty of mathematics.

Additional Resources

For those who want to learn more about absolute value and equations, here are some additional resources:

Khan Academy: Absolute Value Equations
Mathway: Absolute Value Equations
Wolfram Alpha: Absolute Value Equations

These resources provide a wealth of information and practice problems to help you master the concept of absolute value equations.

Discussion

What are your thoughts on absolute value equations? Do you have any questions or topics you'd like to discuss? Share your thoughts in the comments below!

What are the Solutions to the Equation $|x-2|=3$ ?
How to Solve Absolute Value Equations with Variables
The Importance of Absolute Value in Real-World Applications

These articles provide more information on absolute value equations and their applications in real-world scenarios.
Absolute Value Equation Q&A

In our previous article, we explored the solutions to the equation $|y-3|=2$ . We broke down the equation into two separate cases and solved each one to find the solutions. In this article, we'll answer some frequently asked questions about absolute value equations and provide additional insights.

Q: What is the definition of absolute value?

A: The absolute value of a number $x$ , denoted by $|x|$ , is the distance of $x$ from zero on the number line. In other words, it's the magnitude of $x$ without considering its direction.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to rewrite it as two separate equations, one for each case: when the expression inside the absolute value is positive and when it's negative. Then, solve each equation separately to find the solutions.

Q: What are the two cases for an absolute value equation?

A: The two cases for an absolute value equation are:

Case 1: $(y-3)=2$
Case 2: $-(y-3)=2$

These two cases cover all possible scenarios, and solving each one will give you the solutions to the original equation.

Q: How do I know which case to use?

A: To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is positive, use Case 1. If the expression is negative, use Case 2.

Q: Can I use absolute value equations to solve inequalities?

A: Yes, you can use absolute value equations to solve inequalities. However, you need to consider the direction of the inequality and the sign of the expression inside the absolute value.

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

A: Absolute value equations have many real-world applications, including:

Distance and time: Absolute value equations can be used to calculate distances and times in physics and engineering.
Finance: Absolute value equations can be used to calculate stock prices and investment returns.
Computer science: Absolute value equations can be used to optimize algorithms and solve problems in computer science.

Q: How do I graph an absolute value equation?

A: To graph an absolute value equation, you need to plot the two cases separately and then combine them. The graph will be a V-shaped graph with the vertex at the point where the two cases intersect.

Q: Can I use absolute value equations to solve systems of equations?

A: Yes, you can use absolute value equations to solve systems of equations. However, you need to consider the sign of the expression inside the absolute value and the direction of the inequality.

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 the sign of the expression inside the absolute value
Not using the correct case
Not checking the direction of the inequality

Conclusion

In conclusion, absolute value equations are a powerful tool for solving equations and inequalities. By understanding the properties of absolute value and applying the correct techniques, you can solve equations like this one and gain a deeper appreciation for the beauty of mathematics.

Final Thoughts

Additional Resources

For those who want to learn more about absolute value equations, here are some additional resources:

Khan Academy: Absolute Value Equations
Mathway: Absolute Value Equations
Wolfram Alpha: Absolute Value Equations

These resources provide a wealth of information and practice problems to help you master the concept of absolute value equations.

Discussion

What are your thoughts on absolute value equations? Do you have any questions or topics you'd like to discuss? Share your thoughts in the comments below!

What are the Solutions to the Equation $|x-2|=3$ ?
How to Solve Absolute Value Equations with Variables
The Importance of Absolute Value in Real-World Applications

These articles provide more information on absolute value equations and their applications in real-world scenarios.