Solve $|x-9|=0$ for $x$.
A. $x=0$
B. $x=-18$ or $x=9$
C. $x=-9$
D. $x=9$
E. all real numbers
F. no solution
Answer (2)
The absolute value of an expression is zero only when the expression itself is zero.
Set the expression inside the absolute value to zero: x − 9 = 0 .
Solve for x by adding 9 to both sides.
The solution is x = 9 .
Explanation
Understanding the Problem We are given the equation ∣ x − 9∣ = 0 and asked to solve for x . The absolute value of a number is its distance from 0. Therefore, the absolute value of a number is 0 if and only if the number itself is 0.
Setting up the Equation To solve the equation ∣ x − 9∣ = 0 , we need to find the value of x that makes the expression inside the absolute value equal to 0. This is because the absolute value of 0 is 0, and the absolute value of any other number is positive.
Isolating x We set the expression inside the absolute value equal to 0: x − 9 = 0
Solving for x To solve for x , we add 9 to both sides of the equation: x − 9 + 9 = 0 + 9
x = 9
Final Answer Therefore, the solution to the equation ∣ x − 9∣ = 0 is x = 9 .
Examples
Absolute value equations are useful in many real-world scenarios. For example, when manufacturing parts, there is often a tolerance for the dimensions of the parts. If a part is supposed to be 9 cm long, but it can be off by up to 0.05 cm, then the actual length x must satisfy the inequality ∣ x − 9∣ ≤ 0.05 . Solving absolute value equations and inequalities helps engineers and manufacturers ensure that parts meet the required specifications.
The solution to the equation ∣ x − 9∣ = 0 is x = 9 . This is because the absolute value is zero only when the expression inside it is also zero. Therefore, the correct answer is option D: x = 9 .
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