Question

solve and graph the solution set on the number line.
$-6|1-x| < -18$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is . (type your answer in interval notation. use integers or fractions for any numbers in the expression.)
b. the solution set is $\emptyset$.

Explanation:

Step1: Divide both sides by -6, reverse inequality

$\frac{-6|1-x|}{-6} > \frac{-18}{-6}$
Simplify to: $|1-x| > 3$

Step2: Split absolute value inequality

This splits into two cases:
$1-x > 3$ or $1-x < -3$

Step3: Solve first inequality

Subtract 1 from both sides: $-x > 2$
Multiply by -1, reverse inequality: $x < -2$

Step4: Solve second inequality

Subtract 1 from both sides: $-x < -4$
Multiply by -1, reverse inequality: $x > 4$

Step5: Write in interval notation

Combine the two solution sets: $(-\infty, -2) \cup (4, \infty)$

Answer:

A. The solution set is $(-\infty, -2) \cup (4, \infty)$.

(For the number line graph: Draw an open circle at -2 with an arrow pointing left, and an open circle at 4 with an arrow pointing right.)