Solve for $(x): |2x-6| < x$

Analyze the absolute value inequality ∣2 x − 6∣ < x and identify the two cases based on the sign of 2 x − 6 .
Solve the inequality for Case 1 ( x ≥ 3 ), obtaining 3 ≤ x < 6 .
Solve the inequality for Case 2 ( x < 3 ), obtaining 2 < x < 3 .
Combine the solutions from both cases to find the overall solution: 2 < x < 6 ​ .

Explanation

Understanding the Problem We are given the inequality ∣2 x − 6∣ < x . We need to solve for x . The absolute value function is defined as ∣ a ∣ = a if =" 0"> a " >= "0 and ∣ a ∣ = − a if a < 0 . We must have 0"> x > 0 for the inequality to hold, since absolute values are non-negative.

Case 1: x ≥ 3 We need to consider two cases to solve this absolute value inequality:


Case 1: 2 x − 6 ≥ 0 , which means x ≥ 3 . In this case, ∣2 x − 6∣ = 2 x − 6 , so the inequality becomes 2 x − 6 < x .

Solving Case 1 Solving the inequality 2 x − 6 < x :

Subtract x from both sides: 2 x − x − 6 < 0 , which simplifies to x − 6 < 0 .
Add 6 to both sides: x < 6 .
So, in this case, we have x ≥ 3 and x < 6 . Combining these, we get 3 ≤ x < 6 .

Case 2: x < 3 Case 2: 2 x − 6 < 0 , which means x < 3 . In this case, ∣2 x − 6∣ = − ( 2 x − 6 ) = 6 − 2 x , so the inequality becomes 6 − 2 x < x .

Solving Case 2 Solving the inequality 6 − 2 x < x :


Add 2 x to both sides: 6 < x + 2 x , which simplifies to 6 < 3 x .
Divide both sides by 3: 2 < x .
So, in this case, we have x < 3 and 2"> x > 2 . Combining these, we get 2 < x < 3 .

Combining the Solutions Combining the solutions from both cases:

From Case 1, we have 3 ≤ x < 6 .
From Case 2, we have 2 < x < 3 .
Combining these intervals, we get 2 < x < 6 .

Final Answer Therefore, the solution to the inequality ∣2 x − 6∣ < x is 2 < x < 6 .

Examples
Absolute value inequalities can be used to determine the range of acceptable errors in manufacturing. For example, if a machine is supposed to cut a metal rod to 10 cm, but the acceptable error is less than 1 cm, the actual length x must satisfy ∣ x − 10∣ < 1 . Solving this inequality gives the acceptable range of lengths for the rod.

Answered by GinnyAnswer | 2025-07-03

To solve the inequality |2x-6| < x, we split the problem into two cases based on the sign of 2x-6. The solutions are Case 1: 3 ≤ x < 6 and Case 2: 2 < x < 3, leading to the final combined solution: 2 < x < 6.
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Answered by Anonymous | 2025-07-04