$\[ |1-3t| + 2 = 9 \\]Select The Correct Choice Below And Fill In Any Answer Boxes Within Your Choice:A. The Solution Set Is \[$\{ \ \square \ \}\$\]. (Use A Comma To Separate Answers As Needed.)B. The Solution Set Is All Real
Introduction
In this article, we will delve into the world of absolute value equations and solve the given equation |1-3t| + 2 = 9. Absolute value equations are a fundamental concept in algebra, and understanding how to solve them is crucial for success in mathematics. We will break down the solution process step by step, making it easy to follow and understand.
Understanding Absolute Value Equations
Before we dive into solving the equation, let's take a moment to understand what absolute value equations are. An absolute value equation is an equation that contains 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. In other words, the absolute value of a number is always non-negative.
The Given Equation
The given equation is |1-3t| + 2 = 9. Our goal is to solve for the variable t. To do this, we need to isolate the absolute value expression and then solve for t.
Step 1: Isolate the Absolute Value Expression
To isolate the absolute value expression, we need to subtract 2 from both sides of the equation. This will give us |1-3t| = 7.
Step 2: Write Two Separate Equations
Since the absolute value of an expression can be positive or negative, we need to write two separate equations:
1-3t = 7 1-3t = -7
Step 3: Solve the First Equation
Let's solve the first equation: 1-3t = 7. To do this, we need to isolate the variable t. We can do this by subtracting 1 from both sides of the equation, which gives us -3t = 6. Then, we can divide both sides of the equation by -3, which gives us t = -2.
Step 4: Solve the Second Equation
Now, let's solve the second equation: 1-3t = -7. To do this, we need to isolate the variable t. We can do this by subtracting 1 from both sides of the equation, which gives us -3t = -8. Then, we can divide both sides of the equation by -3, which gives us t = 8/3.
Step 5: Check the Solutions
Now that we have found two possible solutions for t, we need to check if they are valid. We can do this by plugging each solution back into the original equation and checking if it is true.
For t = -2, we have |1-3(-2)| + 2 = |1+6| + 2 = 7 + 2 = 9, which is true.
For t = 8/3, we have |1-3(8/3)| + 2 = |1-8| + 2 = |-7| + 2 = 7 + 2 = 9, which is also true.
Conclusion
In conclusion, the solution set for the equation |1-3t| + 2 = 9 is {t = -2, t = 8/3}. This means that the variable t can take on two different values, -2 and 8/3, and both of these values satisfy the original equation.
Final Answer
The final answer is:
A. The solution set is {t = -2, t = 8/3}.
Introduction
In our previous article, we solved the absolute value equation |1-3t| + 2 = 9 and found that the solution set is {t = -2, t = 8/3}. In this article, we will answer some frequently asked questions about absolute value equations and provide additional examples to help solidify your understanding.
Q&A
Q: What is an absolute value equation?
A: An absolute value equation is an equation that contains 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 write two separate equations: one where the expression is positive and one where the expression is negative. Then, you can solve each equation separately and check the solutions to make sure they are valid.
Q: What is the difference between an absolute value equation and a linear equation?
A: An absolute value equation is an equation that contains the absolute value of a variable or expression, while a linear equation is an equation that can be written in the form ax + b = c, where a, b, and c are constants.
Q: Can I use the same method to solve a quadratic equation as I would to solve an absolute value equation?
A: No, the method for solving a quadratic equation is different from the method for solving an absolute value equation. A quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants.
Q: What is the solution set for the equation |2x - 5| = 3?
A: To solve this equation, we need to isolate the absolute value expression and then write two separate equations:
2x - 5 = 3 2x - 5 = -3
Solving each equation separately, we get:
x = 4 x = 1
So, the solution set for the equation |2x - 5| = 3 is {x = 4, x = 1}.
Q: What is the solution set for the equation |x + 2| = 5?
A: To solve this equation, we need to isolate the absolute value expression and then write two separate equations:
x + 2 = 5 x + 2 = -5
Solving each equation separately, we get:
x = 3 x = -7
So, the solution set for the equation |x + 2| = 5 is {x = 3, x = -7}.
Additional Examples
Example 1: |3x - 2| = 4
To solve this equation, we need to isolate the absolute value expression and then write two separate equations:
3x - 2 = 4 3x - 2 = -4
Solving each equation separately, we get:
x = 2 x = 0
So, the solution set for the equation |3x - 2| = 4 is {x = 2, x = 0}.
Example 2: |2x + 1| = 6
To solve this equation, we need to isolate the absolute value expression and then write two separate equations:
2x + 1 = 6 2x + 1 = -6
Solving each equation separately, we get:
x = 2.5 x = -3.5
So, the solution set for the equation |2x + 1| = 6 is {x = 2.5, x = -3.5}.
Conclusion
In conclusion, solving absolute value equations requires a different approach than solving linear or quadratic equations. By isolating the absolute value expression and writing two separate equations, we can find the solution set for the equation. We hope this article has helped you understand how to solve absolute value equations and has provided you with additional examples to help solidify your understanding.
Final Answer
The final answer is:
A. The solution set for the equation |1-3t| + 2 = 9 is {t = -2, t = 8/3}.