Solve the equation.
|x-1|+5=2
The Correct Answer and Explanation is :
The correct answer is: answer is that there are no solutions to the equation ∣x−1∣+5=2|x – 1| + 5 = 2∣x−1∣+5=2.
To solve the equation (|x – 1| + 5 = 2), we first isolate the absolute value term. We start by subtracting 5 from both sides:
[
|x – 1| + 5 – 5 = 2 – 5
]
This simplifies to:
[
|x – 1| = -3
]
Now, we need to analyze the result. The absolute value (|x – 1|) represents the distance between (x) and 1 on the number line. By definition, absolute values cannot be negative. Therefore, (|x – 1| \geq 0) for all values of (x). Since (|x – 1| = -3) results in a negative value, we conclude that there is no solution to the equation.
Explanation:
- Understanding Absolute Values: The absolute value of a number (a), denoted as (|a|), is defined as:
- (a) if (a \geq 0)
- (-a) if (a < 0)
In other words, it always yields a non-negative result. Thus, when we see an equation where the absolute value equals a negative number, we must recognize immediately that it cannot hold true for any real number.
- Process of Solving: The initial equation included the term (|x – 1| + 5). By manipulating the equation to isolate the absolute value, we ensured we were working toward a simpler form. However, upon finding that (|x – 1| = -3), we realized that we had ventured into an impossible situation.
- Conclusion: In summary, the conclusion drawn from the equation (|x – 1| + 5 = 2) is that there are no real solutions. This type of outcome often arises when dealing with absolute value equations that lead to contradictions. In the broader context of mathematics, recognizing when an equation has no solution is as important as finding actual solutions. Thus, the final answer is that there are no solutions to the equation (|x – 1| + 5 = 2).