Solve |2x-2| < 8
{x|-5 < x < 3}
{x|x < -3 or x > 5}
{x|-3 < x < 5}
To solve the inequality |2x-2| < 8, we need to consider two cases: when 2x-2 is positive and when it is negative.
Case 1: 2x-2 is positive. In this case, the absolute value of 2x-2 is the same as 2x-2. Therefore, we can rewrite the inequality as 2x-2 < 8. Solving this inequality:
2x-2 < 8
2x < 10
x < 5
So, in this case, the solution is x < 5.
Case 2: 2x-2 is negative. In this case, the absolute value of 2x-2 is the opposite of 2x-2, so we can rewrite the inequality as -(2x-2) < 8. Simplifying:
-2x+2 < 8
-2x < 6
x > -3
So, in this case, the solution is x > -3.
Combining the solutions from both cases, we have -3 < x < 5.
To solve the inequality |2x-2| < 8, we can break it down into two separate inequalities:
1. 2x-2 < 8
2. 2x-2 > -8
Solving the first inequality:
2x-2 < 8
Add 2 to both sides:
2x < 10
Divide both sides by 2 to isolate x:
x < 5
Solving the second inequality:
2x-2 > -8
Add 2 to both sides:
2x > -6
Divide both sides by 2 to isolate x:
x > -3
Combining the two solutions, we get:
-3 < x < 5
To solve the inequality |2x-2| < 8, we need to break it down into two separate cases.
Case 1: When 2x-2 is non-negative (greater than or equal to 0).
In this case, the absolute value becomes |2x-2| = 2x-2.
So the inequality |2x-2| < 8 becomes 2x-2 < 8.
Now, we can solve this inequality:
2x - 2 < 8
Adding 2 to both sides, we get:
2x < 10
Dividing both sides by 2, we get:
x < 5.
Case 2: When 2x-2 is negative (less than 0).
In this case, the absolute value becomes |2x-2| = -(2x-2) = -2x+2 (Note the negation sign).
So the inequality |2x-2| < 8 becomes -2x + 2 < 8.
Now, we can solve this inequality:
-2x + 2 < 8
Subtracting 2 from both sides, we get:
-2x < 6
Dividing both sides by -2 and reversing the inequality sign, we get:
x > -3.
Thus, combining the results from both cases, we have the solution:
x < 5 and x > -3.
This can also be written as the union of two intervals: (-∞, -3) U (5, ∞).
So the solution to |2x-2| < 8 is {x | x < -3 or x > 5}.