Solve $\left|\frac{x}{5}\right|=0$ for $x$.
A. $x=1$
B. $x=-5,5$
C. $x=-\frac{1}{5}$
D. $x=0$
E. all real numbers
F. no solution
Answer (2)
Recognize that the absolute value of an expression is zero only when the expression itself is zero.
Set 5 x = 0 .
Multiply both sides of the equation by 5 to solve for x .
The solution is x = 0 .
Explanation
Understanding the Problem We are given the equation 5 x = 0 and asked to solve for x . The absolute value of any expression is its distance from zero. Therefore, the absolute value of an expression is zero if and only if the expression itself is zero.
Isolating x Since 5 x = 0 , it must be the case that 5 x = 0 . To solve for x , we multiply both sides of the equation by 5: 5 ⋅ 5 x = 5 ⋅ 0
Finding the Solution This simplifies to x = 0 . Therefore, the solution to the equation 5 x = 0 is x = 0 .
Examples
Absolute value equations are useful in many real-world scenarios. For example, in manufacturing, if you need to produce parts that are a specific size, say 5 cm, you might use absolute value to describe the acceptable tolerance. If the tolerance is 0.1 cm, you could express the acceptable range of sizes as ∣ x − 5∣ ≤ 0.1 , where x is the actual size of the part. This ensures that all parts produced are within an acceptable range of the target size.
The solution to the equation 5 x = 0 is x = 0 . This is because the absolute value is only equal to zero when the expression inside is zero. Thus, the correct option is D: x = 0 .
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