To determine which of the given equations has only one unique solution, let's analyze each one step by step.
Option A: [tex]\( 9 - |2x - 1| = 9 \)[/tex]
1. Isolate the absolute value term:
[tex]\[ 9 - |2x - 1| = 9 \][/tex]
2. Subtract 9 from both sides:
[tex]\[ -|2x - 1| = 0 \][/tex]
3. Multiply both sides by -1:
[tex]\[ |2x - 1| = 0 \][/tex]
4. The absolute value of a number is zero only if the number itself is zero:
[tex]\[ 2x - 1 = 0 \][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x = 1 \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
This equation has only one unique solution: [tex]\( x = \frac{1}{2} \)[/tex].
Option B: [tex]\( -4|x| + 1 = -3 \)[/tex]
1. Isolate the absolute value term:
[tex]\[ -4|x| + 1 = -3 \][/tex]
2. Subtract 1 from both sides:
[tex]\[ -4|x| = -4 \][/tex]
3. Divide both sides by -4:
[tex]\[ |x| = 1 \][/tex]
4. The absolute value of [tex]\( x \)[/tex] equals 1, so:
[tex]\[ x = 1 \text{ or } x = -1 \][/tex]
This equation has two solutions: [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
Option C: [tex]\( |x - 2| - 7 = 7 \)[/tex]
1. Isolate the absolute value term:
[tex]\[ |x - 2| - 7 = 7 \][/tex]
2. Add 7 to both sides:
[tex]\[ |x - 2| = 14 \][/tex]
3. The absolute value of [tex]\( x - 2 \)[/tex] equals 14, so:
[tex]\[ x - 2 = 14 \text{ or } x - 2 = -14 \][/tex]
4. Solve for [tex]\( x \)[/tex] in both cases:
[tex]\[ x = 16 \text{ or } x = -12 \][/tex]
This equation has two solutions: [tex]\( x = 16 \)[/tex] and [tex]\( x = -12 \)[/tex].
Option D: [tex]\( |3 - x| = 3 \)[/tex]
1. The absolute value of [tex]\( 3 - x \)[/tex] equals 3, so:
[tex]\[ 3 - x = 3 \text{ or } 3 - x = -3 \][/tex]
2. Solve for [tex]\( x \)[/tex] in both cases:
[tex]\[ 3 - x = 3 \Rightarrow -x = 0 \Rightarrow x = 0 \][/tex]
[tex]\[ 3 - x = -3 \Rightarrow -x = -6 \Rightarrow x = 6 \][/tex]
This equation has two solutions: [tex]\( x = 0 \)[/tex] and [tex]\( x = 6 \)[/tex].
From our analysis, only Option A has one unique solution. Therefore, the correct answer is:
A. [tex]\( 9 - |2x - 1| = 9 \)[/tex]
