Solve for (x): [|5x-6|]
Answer (2)
The solution involves solving the equation [ ∣5 x − 6∣ ] = n for x , where n is a non-negative integer. We split the problem into two cases based on the sign of 5 x − 6 . In the first case, 5 x − 6 ≥ 0 , we find that 5 n + 6 ≤ x < 5 n + 7 . In the second case, 5 x − 6 < 0 , we find that 5 5 − n < x ≤ 5 6 − n . Combining these cases, the solution is given by the intervals 5 n + 6 ≤ x < 5 n + 7 or 5 5 − n < x ≤ 5 6 − n . The final answer is 5 n + 6 ≤ x < 5 n + 7 or 5 5 − n < x ≤ 5 6 − n
Explanation
Understanding the Problem We are asked to solve for x in the expression [ ∣5 x − 6∣ ] . The notation [ y ] represents the greatest integer less than or equal to y , also known as the floor function. The expression ∣5 x − 6∣ represents the absolute value of 5 x − 6 . Since the right-hand side of the equation is not specified, I assume the problem is to find the values of x for which [ ∣5 x − 6∣ ] equals a certain integer n .
Setting up the Cases Let [ ∣5 x − 6∣ ] = n , where n is an integer. Then n ≤ ∣5 x − 6∣ < n + 1 . We consider two cases: 5 x − 6 ≥ 0 and 5 x − 6 < 0 .
Analyzing Case 1 Case 1: 5 x − 6 ≥ 0 , which means x ≥ 5 6 . In this case, ∣5 x − 6∣ = 5 x − 6 , so n ≤ 5 x − 6 < n + 1 . Adding 6 to all sides gives n + 6 ≤ 5 x < n + 7 . Dividing by 5 gives 5 n + 6 ≤ x < 5 n + 7 . Since x ≥ 5 6 , we must have 5 n + 6 ≥ 5 6 , which means n + 6 ≥ 6 , so n ≥ 0 .
Analyzing Case 2 Case 2: 5 x − 6 < 0 , which means x < 5 6 . In this case, ∣5 x − 6∣ = − ( 5 x − 6 ) = 6 − 5 x , so n ≤ 6 − 5 x < n + 1 . Subtracting 6 from all sides gives n − 6 ≤ − 5 x < n − 5 . Multiplying by -1 and reversing the inequality signs gives 5 − n < 5 x ≤ 6 − n . Dividing by 5 gives 5 5 − n < x ≤ 5 6 − n . Since x < 5 6 , we must have 5 6 − n ≤ 5 6 , which means 6 − n ≤ 6 , so − n ≤ 0 , which means n ≥ 0 .
Combining the Results Combining the two cases, for a given non-negative integer n , the solution is 5 n + 6 ≤ x < 5 n + 7 or 5 5 − n < x ≤ 5 6 − n .
Final Answer Therefore, the solution for x is given by the intervals 5 n + 6 ≤ x < 5 n + 7 or 5 5 − n < x ≤ 5 6 − n , where n is a non-negative integer.
Examples
Absolute value and floor functions are used in computer science for error correction and data analysis. For example, when dealing with sensor readings, absolute values can help quantify the magnitude of deviations from a target value, while floor functions can be used to discretize continuous data into integer bins for analysis. This combination allows for robust and simplified data processing in various applications, such as robotics and environmental monitoring.
To solve ∣5 x − 6∣ for values of x , we analyze two cases depending on whether 5 x − 6 is positive or negative. Ultimately, we find that for non-negative integers n , the possible intervals for x can be expressed mathematically. The resulting intervals illustrate the possible solutions to the absolute value equation.
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