How do you solve and write the following in interval notation: #| 2x – 4 | ≤ 12#?

Question

Abigail Allen · Accepted Answer

See the entire solution process below:

The absolute value function takes any term, negative or positive, and transforms it to its positive form. Therefore, you need to solve the term inside the absolute value for both the positive and negative version of what it is equated to. #-12 <= 2x - 4 <= 12# #-12 + color(red)(4) <= 2x - 4 + color(red)(4) <= 12 + color(red)(4)# #-8 <= 2x - 0 <= 16# #-8 <= 2x <= 16# #-8/color(red)(2) <= (2x)/color(red)(2) <= 16/color(red)(2)# #-4 <= (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) <= 8# #-4 <= x <= 8# Or #x >= -4# and #x <= 8# Or, in interval notation #[-4, 8]#

Ethan Adkins · Answer

undefined To solve the inequality |2x - 4| ≤ 12 and write it in interval notation:

1. Solve the inequality |2x - 4| ≤ 12:
   - First, solve the equation 2x - 4 = 12 to find the upper bound.
   - Then, solve the equation 2x - 4 = -12 to find the lower bound.
   - The solutions to these equations represent the intervals where the inequality holds true.

2. Write the solutions in interval notation:
   - Use square brackets [ ] to denote inclusive boundaries and round parentheses ( ) to denote exclusive boundaries.
   - Combine the intervals into a single interval if there are multiple solutions.