Solve each inequality and express the solution set in interval notation.
a. 3 < |(x-3)/2| < 5
b. 5 ≥ |(x+2)/3| > 1
ans.: (a)(9,-3) (-7,13)
(b) [-17,13] (1,-5)
To solve inequalities and express the solution set in interval notation, we need to follow a few steps. Let's go through each inequality:
a. 3 < |(x-3)/2| < 5
Step 1: Remove the absolute value by creating two separate inequalities:
(x-3)/2 > 3 and (x-3)/2 < -3
OR
(x-3)/2 > 5 and (x-3)/2 < -5
Step 2: Solve each inequality separately:
(x-3)/2 > 3
x - 3 > 6
x > 9
(x-3)/2 < -3
x - 3 < -6
x < -3
(x-3)/2 > 5
x - 3 > 10
x > 13
(x-3)/2 < -5
x - 3 < -10
x < -7
Step 3: Combine the solutions to get the final solution set:
(-∞, -7) U (9, ∞)
b. 5 ≥ |(x+2)/3| > 1
Step 1: Remove the absolute value by creating two separate inequalities:
(x+2)/3 ≤ 5 and (x+2)/3 > 1
Step 2: Solve each inequality separately:
(x+2)/3 ≤ 5
x + 2 ≤ 15
x ≤ 13
(x+2)/3 > 1
x + 2 > 3
x > 1
Step 3: Combine the solutions to get the final solution set:
[1, 13]
Therefore, the solution sets in interval notation are:
(a) (9, ∞) U (-∞, -7)
(b) [1, 13]