Which number line represents the solutions to $|x-5|=1$?
Answer (1)
The equation ∣ x − 5∣ = 1 means that x − 5 can be either 1 or -1.
Solve x − 5 = 1 to get x = 6 .
Solve x − 5 = − 1 to get x = 4 .
The solutions are x = 4 and x = 6 , which should be represented on a number line with closed circles at 4 and 6. 4 , 6
Explanation
Understanding the Problem We are given the equation ∣ x − 5∣ = 1 and asked to find the number line that represents its solutions. This is an absolute value equation, which means we need to consider two cases: when the expression inside the absolute value is positive or zero, and when it is negative.
Solving Case 1 Case 1: x − 5 = 1 . To solve for x , we add 5 to both sides of the equation: x − 5 + 5 = 1 + 5 x = 6
Solving Case 2 Case 2: x − 5 = − 1 . To solve for x , we add 5 to both sides of the equation: x − 5 + 5 = − 1 + 5 x = 4
Finding the Solutions Therefore, the solutions to the equation ∣ x − 5∣ = 1 are x = 4 and x = 6 . The number line representing these solutions will have closed circles (or dots) at 4 and 6.
Examples
Absolute value equations are useful in many real-world scenarios, such as determining tolerances in manufacturing. For example, if a machine part is designed to be 5 cm long, but a tolerance of 1 mm (0.1 cm) is allowed, the actual length x must satisfy ∣ x − 5∣ l e 0.1 . This means the length can be between 4.9 cm and 5.1 cm. Understanding absolute value equations helps engineers ensure parts meet the required specifications.
