Solve the absolute value equation. [tex]|x|=-1[/tex]
Answer (2)
The absolute value of a number is always non-negative.
The equation ∣ x ∣ = − 1 implies the absolute value of x is -1, which is impossible.
Therefore, the equation has no solution.
The final answer is no solution .
Explanation
Understanding the Problem The problem is to solve the absolute value equation ∣ x ∣ = − 1 . We need to determine which of the given options is the correct solution.
Understanding Absolute Value Recall that the absolute value of a real number x , denoted by ∣ x ∣ , is defined as the distance of x from 0 on the number line. Therefore, ∣ x ∣ is always non-negative, i.e., = 0"> ∣ x ∣" >= 0 for any real number x .
Applying the Definition In the given equation, ∣ x ∣ = − 1 , the absolute value of x is equal to − 1 . However, since the absolute value of any real number is always non-negative, it cannot be equal to a negative number like − 1 .
Finding the Solution Therefore, there is no real number x that satisfies the equation ∣ x ∣ = − 1 .
Examples
Absolute value equations are useful in many real-world scenarios. For example, when measuring errors in manufacturing, we often care about the magnitude of the error, not its direction. If a machine is supposed to cut a metal rod to 10 cm, and it cuts one to 9.8 cm and another to 10.2 cm, the absolute value helps us quantify the error in both cases as 0.2 cm. Similarly, in navigation, absolute values are used to calculate distances, regardless of the direction of travel. Understanding absolute values helps in quality control, navigation, and any situation where magnitude is more important than direction.
The equation ∣ x ∣ = − 1 has no solution because the absolute value of any real number is always non-negative. Therefore, it cannot equal a negative number like − 1 . Thus, the answer is no solution .
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