Question

which equation is an example of an absolute value equation that has no solutions?\\(\vert 81x + 5\vert - 22 = -10\\)\\(\vert 3x + 7\vert - 16 = -3\\)\\(\vert 14x - 16\vert - 0.5 = 0\\)\\(\vert 14x - 7\vert + 1 = -3\\)

Explanation:

Step1: Recall absolute value property

The absolute value of any real number, $|A|$, satisfies $|A| \geq 0$ for all real $A$.

Step2: Analyze each option

Option1: Isolate absolute value

$|81x + 5| = -10 + 22 = 12$. Since $12 \geq 0$, this has solutions.

Option2: Isolate absolute value

$|3x + 7| = -3 + 15 = 12$. Since $12 \geq 0$, this has solutions.

Option3: Isolate absolute value

$|14x - 16| = 0.5$. Since $0.5 \geq 0$, this has solutions.

Option4: Isolate absolute value

$|14x - 7| = -3 - 1 = -4$. Since $-4 < 0$, no real number can satisfy this, so there are no solutions.

Answer:

$\boldsymbol{|14x - 7| + 1 = -3}$