which equation has only one solution?\n|x - 5|=-1\n|-6 - 2x| = 8\n|5x + 10| = 10\n|-6x + 3| = 0

Explanation:

Step1: Recall absolute - value property

The absolute - value of a number (|a|) is always non - negative, i.e., (|a|\geq0) for all real numbers (a).

Step2: Analyze (|x - 5|=-1)

Since the absolute - value of any real number cannot be negative, the equation (|x - 5|=-1) has no solutions.

Step3: Analyze (|-6 - 2x| = 8)

By the definition of absolute value, if (|u|=c) ((c>0)), then (u = c) or (u=-c). So, (-6 - 2x=8) or (-6 - 2x=-8). For (-6 - 2x=8), we have (-2x=8 + 6=14), then (x=-7). For (-6 - 2x=-8), we have (-2x=-8 + 6=-2), then (x = 1). This equation has two solutions.

Step4: Analyze (|5x + 10| = 10)

If (|u|=c) ((c>0)), then (u = c) or (u=-c). So, (5x+10 = 10) or (5x+10=-10). For (5x+10 = 10), (5x=0), then (x = 0). For (5x+10=-10), (5x=-20), then (x=-4). This equation has two solutions.

Step5: Analyze (|-6x + 3| = 0)

By the definition of absolute value, if (|u| = 0), then (u = 0). So, (-6x+3 = 0). Subtract 3 from both sides: (-6x=-3). Divide both sides by (-6): (x=\frac{1}{2}). This equation has only one solution.

Answer:

(|-6x + 3| = 0)