Answer :
Let's analyze each equation and inequality to determine if they have a solution set that is not empty.
### 1. \( 2 |x-8| + 6 < 4 \)
First, isolate the absolute value term:
[tex]\[ 2 |x-8| + 6 < 4 \][/tex]
Subtract 6 from both sides:
[tex]\[ 2 |x-8| < -2 \][/tex]
Divide both sides by 2:
[tex]\[ |x-8| < -1 \][/tex]
The absolute value of any expression is always non-negative, meaning it cannot be less than -1. Therefore, this inequality has no solution.
### 2. \( |2x-1| + 5 = 4 \)
First, isolate the absolute value term:
[tex]\[ |2x-1| + 5 = 4 \][/tex]
Subtract 5 from both sides:
[tex]\[ |2x-1| = -1 \][/tex]
The absolute value of any expression is always non-negative, meaning it cannot equal -1. Therefore, this equation has no solution.
### 3. \( -2 |x-2| + 3 > 9 \)
First, isolate the absolute value term:
[tex]\[ -2 |x-2| + 3 > 9 \][/tex]
Subtract 3 from both sides:
[tex]\[ -2 |x-2| > 6 \][/tex]
Divide both sides by -2 (and remember to reverse the inequality):
[tex]\[ |x-2| < -3 \][/tex]
The absolute value of any expression is always non-negative, meaning it cannot be less than -3. Therefore, this inequality has no solution.
### 4. \( |3x-1| - 5 = -1 \)
First, isolate the absolute value term:
[tex]\[ |3x-1| - 5 = -1 \][/tex]
Add 5 to both sides:
[tex]\[ |3x-1| = 4 \][/tex]
The absolute value equation \( |3x-1| = 4 \) can be split into two cases:
[tex]\[ 3x-1 = 4 \quad \text{or} \quad 3x-1 = -4 \][/tex]
Case 1: \( 3x-1 = 4 \)
[tex]\[ 3x = 5 \][/tex]
[tex]\[ x = \frac{5}{3} \][/tex]
Case 2: \( 3x-1 = -4 \)
[tex]\[ 3x = -3 \][/tex]
[tex]\[ x = -1 \][/tex]
Thus, the solutions to the equation \( |3x-1| = 4 \) are \( x = \frac{5}{3} \) and \( x = -1 \), which means this equation has a solution set that is not empty.
### Conclusion
The absolute value equation or inequality with a solution set that is NOT the empty set is:
[tex]\[ |3x-1| - 5 = -1 \][/tex]
### 1. \( 2 |x-8| + 6 < 4 \)
First, isolate the absolute value term:
[tex]\[ 2 |x-8| + 6 < 4 \][/tex]
Subtract 6 from both sides:
[tex]\[ 2 |x-8| < -2 \][/tex]
Divide both sides by 2:
[tex]\[ |x-8| < -1 \][/tex]
The absolute value of any expression is always non-negative, meaning it cannot be less than -1. Therefore, this inequality has no solution.
### 2. \( |2x-1| + 5 = 4 \)
First, isolate the absolute value term:
[tex]\[ |2x-1| + 5 = 4 \][/tex]
Subtract 5 from both sides:
[tex]\[ |2x-1| = -1 \][/tex]
The absolute value of any expression is always non-negative, meaning it cannot equal -1. Therefore, this equation has no solution.
### 3. \( -2 |x-2| + 3 > 9 \)
First, isolate the absolute value term:
[tex]\[ -2 |x-2| + 3 > 9 \][/tex]
Subtract 3 from both sides:
[tex]\[ -2 |x-2| > 6 \][/tex]
Divide both sides by -2 (and remember to reverse the inequality):
[tex]\[ |x-2| < -3 \][/tex]
The absolute value of any expression is always non-negative, meaning it cannot be less than -3. Therefore, this inequality has no solution.
### 4. \( |3x-1| - 5 = -1 \)
First, isolate the absolute value term:
[tex]\[ |3x-1| - 5 = -1 \][/tex]
Add 5 to both sides:
[tex]\[ |3x-1| = 4 \][/tex]
The absolute value equation \( |3x-1| = 4 \) can be split into two cases:
[tex]\[ 3x-1 = 4 \quad \text{or} \quad 3x-1 = -4 \][/tex]
Case 1: \( 3x-1 = 4 \)
[tex]\[ 3x = 5 \][/tex]
[tex]\[ x = \frac{5}{3} \][/tex]
Case 2: \( 3x-1 = -4 \)
[tex]\[ 3x = -3 \][/tex]
[tex]\[ x = -1 \][/tex]
Thus, the solutions to the equation \( |3x-1| = 4 \) are \( x = \frac{5}{3} \) and \( x = -1 \), which means this equation has a solution set that is not empty.
### Conclusion
The absolute value equation or inequality with a solution set that is NOT the empty set is:
[tex]\[ |3x-1| - 5 = -1 \][/tex]
