i need help with these problems please
[6-2x]<4
which is the numeral set for [2x-]<5
1.[x]-4<x<1]
2.{x]x<-4orx>1}
3.{x]-1<x<4}
4.{x]x<-1orx>4}
To solve these inequality problems, we need to analyze each one separately.
1. [6-2x] < 4
To solve this inequality, we will remove the absolute value brackets and create two cases: one when the expression inside the absolute value is positive, and another when it is negative.
Case 1: [6-2x] < 4, when 6 - 2x ≥ 0:
First, solve for 6 - 2x = 0:
6 - 2x = 0
-2x = -6
x = 3
Since this is a strict inequality (<), the solution is x < 3.
Case 2: [6-2x] < 4, when 6 - 2x < 0:
In this case, we flip the inequality sign and solve for 6 - 2x > -4:
6 - 2x > -4
-2x > -10
x < 5
Combining both cases, we have -∞ < x < 3 and x < 5.
2. [2x-] < 5
Following the same steps as before, we set up two cases based on whether the expression inside the absolute value is positive or negative.
Case 1: [2x-] < 5, when 2x - ≥ 0:
First, solve for 2x - = 0:
2x - = 0
- = 2x
As there is no solution for this case (the expression inside the absolute value never equals zero), we move on to Case 2.
Case 2: [2x-] < 5, when 2x - < 0:
Flipping the inequality sign, we solve for 2x - > -5:
2x - > -5
2x > -5
x > -2.5
Since Case 1 led to no solutions, we only have x > -2.5 as the solution.
From the provided options, the correct numeral set representation for this inequality would be 4. {x]x < -1 or x > 4}.