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Which equation has no solution?

A. [tex]\(|4x - 2| = -6\)[/tex]

B. [tex]\(|-2 - x| = 9\)[/tex]

C. [tex]\(|3x + 6| = 6\)[/tex]

D. [tex]\(|-2x + 8| = 0\)[/tex]

Sagot :

GinnyAnswer
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]
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