Final answer:
The solution to the inequality 2+|x-4|>6 in interval notation is (−∞,0)∪(8,+∞)
To solve the inequality 2+∣x−4∣>6 and express the solution in interval notation, follow these steps:
Subtract 2 from both sides of the inequality:∣x−4∣>6−2
Simplify: ∣x−4∣>4
To solve the absolute value inequality
∣x−4∣>4, set up two separate inequalities:
a. x−4>4 (when x−4 is positive)
b. −(x−4)>4 (when x−4 is negative)
Solve each inequality separately:
a. x−4>4
Add 4 to both sides:x>8
b.−(x−4)>4
Distribute the negative sign:−x+4>4
Subtract 4 from both sides:−x>0
Multiply both sides by -1 (this reverses the inequality):x<0
Combine the solutions:
The solution for x is x<0 or x>8.
Express the solution in interval notation:(−∞,0)∪(8,+∞)