Find The Real Solutions Of The Equation. ∣ 1 − 5 T ∣ + 2 = 18 |1-5t| + 2 = 18 ∣1 − 5 T ∣ + 2 = 18 Select The Correct Choice Below And Fill In Any Answer Boxes Within Your Choice.A. The Solution Set Is { □ } \{\ \square \ \} { □ } .(Simplify Your Answer. Use A Comma To Separate Answers
Understanding the Absolute Value Equation
When dealing with absolute value equations, we need to consider two cases: one where the expression inside the absolute value is positive, and another where it is negative. In this case, we have the equation |1-5t| + 2 = 18. Our goal is to find the real solutions for the variable t.
Case 1: 1-5t ≥ 0
In this case, the expression inside the absolute value is non-negative. We can rewrite the equation as 1-5t + 2 = 18. Simplifying this equation, we get -5t + 3 = 18. Subtracting 3 from both sides, we have -5t = 15. Dividing both sides by -5, we find that t = -3.
Case 2: 1-5t < 0
In this case, the expression inside the absolute value is negative. We can rewrite the equation as -(1-5t) + 2 = 18. Simplifying this equation, we get -1 + 5t + 2 = 18. Combining like terms, we have 5t + 1 = 18. Subtracting 1 from both sides, we get 5t = 17. Dividing both sides by 5, we find that t = 17/5.
Checking the Solutions
We need to check if the solutions we found satisfy the original equation. For t = -3, we have |1-5(-3)| + 2 = |1 + 15| + 2 = 16 + 2 = 18, which is true. For t = 17/5, we have |1-5(17/5)| + 2 = |1 - 17| + 2 = |-16| + 2 = 16 + 2 = 18, which is also true.
Conclusion
The real solutions of the equation |1-5t| + 2 = 18 are t = -3 and t = 17/5.
Final Answer
The final answer is:
Understanding the Absolute Value Equation
When dealing with absolute value equations, we need to consider two cases: one where the expression inside the absolute value is positive, and another where it is negative. In this case, we have the equation |1-5t| + 2 = 18. Our goal is to find the real solutions for the variable t.
Q: What is the first step in solving an absolute value equation?
A: The first step in solving an absolute value equation is to consider two cases: one where the expression inside the absolute value is positive, and another where it is negative.
Q: How do we rewrite the equation when the expression inside the absolute value is non-negative?
A: When the expression inside the absolute value is non-negative, we can rewrite the equation by removing the absolute value sign and simplifying the expression.
Q: How do we rewrite the equation when the expression inside the absolute value is negative?
A: When the expression inside the absolute value is negative, we can rewrite the equation by multiplying the expression inside the absolute value by -1 and removing the absolute value sign.
Q: What is the next step after rewriting the equation in both cases?
A: After rewriting the equation in both cases, we need to solve for the variable t in each case.
Q: How do we check if the solutions we found satisfy the original equation?
A: We need to plug the solutions we found back into the original equation to check if they are true.
Q: What are the real solutions of the equation |1-5t| + 2 = 18?
A: The real solutions of the equation |1-5t| + 2 = 18 are t = -3 and t = 17/5.
Q: Why is it important to consider both cases when solving an absolute value equation?
A: It is important to consider both cases because the expression inside the absolute value can be either positive or negative, and we need to account for both possibilities.
Q: What is the final answer to the equation |1-5t| + 2 = 18?
A: The final answer is:
Common Mistakes to Avoid
- Not considering both cases when solving an absolute value equation
- Not rewriting the equation correctly in both cases
- Not checking if the solutions satisfy the original equation
- Not accounting for the possibility of multiple solutions
Tips for Solving Absolute Value Equations
- Always consider both cases when solving an absolute value equation
- Rewrite the equation correctly in both cases
- Check if the solutions satisfy the original equation
- Account for the possibility of multiple solutions
Conclusion
Solving absolute value equations requires careful consideration of both cases and attention to detail. By following the steps outlined in this article, you can find the real solutions of the equation |1-5t| + 2 = 18 and avoid common mistakes.