Given the equations:
1. [tex]\( |4x - 2| = -6 \)[/tex]
2. [tex]\( |-2 - x| = 9 \)[/tex]
3. [tex]\( |3x + 6| = 6 \)[/tex]
4. [tex]\( |-2x + 8| = 0 \)[/tex]
We'll solve each equation and identify if any have no solutions.
### Equation 1: [tex]\( |4x - 2| = -6 \)[/tex]
The absolute value expression [tex]\(|4x - 2|\)[/tex] represents a distance, which is always non-negative. Therefore:
[tex]\[ |4x - 2| \geq 0 \][/tex]
Since [tex]\(-6\)[/tex] is negative, this equation cannot be true for any value of [tex]\(x\)[/tex]. Thus, this equation has no solution.
### Equation 2: [tex]\( |-2 - x| = 9 \)[/tex]
We solve for [tex]\(x\)[/tex] using the properties of absolute values. This equation has two cases:
[tex]\[ |-2 - x| = 9 \Rightarrow -2 - x = 9 \quad \text{or} \quad -2 - x = -9 \][/tex]
Case 1:
[tex]\[ -2 - x = 9 \][/tex]
[tex]\[ -x = 11 \][/tex]
[tex]\[ x = -11 \][/tex]
Case 2:
[tex]\[ -2 - x = -9 \][/tex]
[tex]\[ -x = -7 \][/tex]
[tex]\[ x = 7 \][/tex]
Both cases provide valid solutions ([tex]\(x = -11\)[/tex] and [tex]\(x = 7\)[/tex]), so this equation has solutions.
### Equation 3: [tex]\( |3x + 6| = 6 \)[/tex]
We solve for [tex]\(x\)[/tex] using the properties of absolute values. This equation also has two cases:
[tex]\[ |3x + 6| = 6 \Rightarrow 3x + 6 = 6 \quad \text{or} \quad 3x + 6 = -6 \][/tex]
Case 1:
[tex]\[ 3x + 6 = 6 \][/tex]
[tex]\[ 3x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
Case 2:
[tex]\[ 3x + 6 = -6 \][/tex]
[tex]\[ 3x = -12 \][/tex]
[tex]\[ x = -4 \][/tex]
Both cases provide valid solutions ([tex]\(x = 0\)[/tex] and [tex]\(x = -4\)[/tex]), so this equation has solutions.
### Equation 4: [tex]\( |-2x + 8| = 0 \)[/tex]
The absolute value expression equals zero only when the quantity inside it is zero:
[tex]\[ |-2x + 8| = 0 \Rightarrow -2x + 8 = 0 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ -2x + 8 = 0 \][/tex]
[tex]\[ -2x = -8 \][/tex]
[tex]\[ x = 4 \][/tex]
This provides a valid solution ([tex]\(x = 4\)[/tex]), so this equation has a solution.
### Conclusion
The only equation that has no solution is:
[tex]\[ |4x - 2| = -6 \][/tex]
