1. Solve each inequality and express the solution set in interval notation.
a. 3 < |(x-3)/2| < 5
b. 5 ≥ |(x+2)/3| > 1
2. 4 < 2x-8 ≤ 10
ans.: (a)(9,-3) (-7,13)
(b) [-17,13] (1,-5)
2. (6,9]
type in your functions here, and see whether you can see how to arrive at the solutions.
http://www.wolframalpha.com/input/?i=%7C(x-3)%2F2%7C
To solve each inequality and express the solution set in interval notation, we need to follow these steps:
1. Isolate the absolute value expression by removing the absolute value signs.
2. Set the expression inside the absolute value greater than or equal to zero and solve for the intervals.
3. Set the expression inside the absolute value less than the given inequality value and solve for the intervals.
4. Combine the intervals from steps 2 and 3 to find the final solution set.
Let's solve each inequality step by step:
a. 3 < |(x-3)/2| < 5
Step 1: Remove the absolute value signs:
-5 < (x-3)/2 < 5
Step 2: Set the expression inside the absolute value greater than or equal to zero:
(x-3)/2 ≥ 0
x - 3 ≥ 0
x ≥ 3
Step 3: Set the expression inside the absolute value less than the given inequality values:
-5 < (x-3)/2 < 5
For (x-3)/2 < 5:
x - 3 < 10
x < 13
For (x-3)/2 > -5:
x - 3 > -10
x > -7
Step 4: Combine the intervals:
The final solution set in interval notation is (-7, 13).
b. 5 ≥ |(x+2)/3| > 1
Step 1: Remove the absolute value signs:
1 ≤ (x+2)/3 ≤ 5
Step 2: Set the expression inside the absolute value greater than or equal to zero:
(x+2)/3 ≥ 0
x + 2 ≥ 0
x ≥ -2
Step 3: Set the expression inside the absolute value less than the given inequality values:
1 ≤ (x+2)/3 ≤ 5
For (x+2)/3 > 1:
x + 2 > 3
x > 1
For (x+2)/3 < 5:
x + 2 < 15
x < 13
Step 4: Combine the intervals:
The final solution set in interval notation is (-2, 13].
For the third question:
2. 4 < 2x-8 ≤ 10
Step 1: Simplify the inequality:
4 < 2x - 8 ≤ 10
Step 2: Solve individually for each inequality:
For 2x - 8 > 4:
2x > 12
x > 6
For 2x - 8 ≤ 10:
2x ≤ 18
x ≤ 9
Step 3: Combine the intervals:
The final solution set in interval notation is (6, 9].