Question

solve and graph the solution set on a number line. $left|2 - \frac{x}{2}
ight| > 1$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is $(-\infty,2)\cup(6,\infty)$. (type your answer in interval notation.)
b. the solution set is $\emptyset$.
choose the correct graph below.
a. number line graph with open circles at 2 and 6, arrows pointing left from 2 and right from 6
b. number line graph with open circles at 2 and 6, line segment between them
c. number line graph with closed circles at 2 and 6, arrows pointing left from 2 and right from 6
d. the graph contains no points.

Explanation:

Step1: Apply absolute value rule

For $|A|>1$, $A>1$ or $A<-1$.
So $2-\frac{x}{2}>1$ or $2-\frac{x}{2}<-1$

Step2: Solve first inequality

Isolate $x$:
$2-\frac{x}{2}>1$
$-\frac{x}{2}>1-2$
$-\frac{x}{2}>-1$
Multiply by $-2$ (reverse inequality):
$x<2$

Step3: Solve second inequality

Isolate $x$:
$2-\frac{x}{2}<-1$
$-\frac{x}{2}<-1-2$
$-\frac{x}{2}<-3$
Multiply by $-2$ (reverse inequality):
$x>6$

Step4: Match to interval notation

Combine solutions: $(-\infty,2)\cup(6,\infty)$

Step5: Match to number line

Open circles at 2 and 6, arrows pointing left from 2 and right from 6.

Answer:

Solution Set:

$(-\infty,2)\cup(6,\infty)$

Correct Choices:

A. The solution set is $(-\infty,2)\cup(6,\infty)$. (Type your answer in interval notation.)
A. (The number line with open circles at 2 and 6, shading to the left of 2 and right of 6)