To solve the inequality involving absolute value, [tex]\[3 + 2|x - 1| < 9,\][/tex] we need to break it down into two separate inequalities, removing the absolute value. The absolute value inequality [tex]\( |x - 1| < k \)[/tex] can be written as two simultaneous inequalities:
[tex]\[ -k < x - 1 < k. \][/tex]
Here are the detailed steps to solve the given inequality:
1. Start with the inequality:
[tex]\[ 3 + 2|x - 1| < 9. \][/tex]
2. Isolate the absolute value expression:
[tex]\[ 2|x - 1| < 6 \][/tex]
(Subtract 3 from both sides.)
3. Divide both sides by 2 to simplify:
[tex]\[ |x - 1| < 3. \][/tex]
4. Now, break down the absolute value inequality into two separate inequalities:
[tex]\[ -3 < x - 1 < 3. \][/tex]
5. Solve each part of the inequality separately:
- For the left part: [tex]\[ -3 < x - 1 \][/tex]
Add 1 to both sides:
[tex]\[ -2 < x \][/tex]
Or equivalently: [tex]\[ x > -2. \][/tex]
- For the right part: [tex]\[ x - 1 < 3 \][/tex]
Add 1 to both sides:
[tex]\[ x < 4. \][/tex]
6. Combining both results, we have:
[tex]\[ -2 < x < 4. \][/tex]
This means the solution to the inequality [tex]\[ 3 + 2|x - 1| < 9 \][/tex] is found using the inequalities:
[tex]\[ x - 1 < 3 \quad \text{and} \quad x - 1 > -3 \][/tex]
Answering the original question "Select all the correct answers", the two inequalities you can use are:
[tex]\[ x - 1 < 3 \quad \text{and} \quad x - 1 > -3 \][/tex]
So, the correct choices are:
[tex]\[ x - 1 < 3 \][/tex]
[tex]\[ x - 1 > -3 \][/tex]
