What is |11-2x|-6>11 in set and interval notation?
To solve the inequality |11-2x| - 6 > 11, we need to isolate the absolute value expression and solve for x. Let's break down the steps:
Step 1: Remove the outer absolute value brackets.
We have two cases to consider: 11-2x could be positive or negative. To do this, we write two separate inequalities without the absolute value:
11 - 2x - 6 > 11 (when 11-2x > 0)
and
-(11 - 2x) - 6 > 11 (when 11-2x < 0)
Step 2: Solve the inequalities separately.
Case 1: 11 - 2x - 6 > 11 (when 11-2x > 0)
Simplifying the inequality:
5 - 2x > 11
-2x > 6 (subtracting 5 from both sides)
x < -3 (dividing both sides by -2 and flipping the inequality sign)
Case 2: -(11 - 2x) - 6 > 11 (when 11-2x < 0)
Simplifying the inequality:
-11 + 2x - 6 > 11
2x > 28
x > 14
Now we have the solutions for each case: x < -3 and x > 14. However, we need to consider the intersection of these two solution sets.
For x < -3 and x > 14 to be simultaneous, there must be no intersecting values. In other words, there are no real values that satisfy both inequalities simultaneously. Therefore, the solution set is empty.
In set notation, we express an empty solution set as an empty set: {}.
In interval notation, an empty solution set is represented by the symbol ∅.
Hence, the solution in set notation is {} and in interval notation is ∅.