Question

solve the absolute - value inequality. graph the solution set. |1 - 2x| - 9 < - 4
select the correct choice and, if necessary, fill in the answer box within your choice
a. the solution set in interval notation is (type your answer in interval notation. use integers or fractions for any numbers in the expression.)
b. the solution is the empty set.
graph the solution set. choose the correct answer
a. graph options with number - lines
b. graph options with number - lines
c. graph options with number - lines
d. graph options with number - lines
e. graph options with number - lines
f. graph options with number - lines

Explanation:

Step1: Isolate the absolute - value expression

Add 9 to both sides of the inequality $|1 - 2x|-9\lt - 4$:
$|1 - 2x|\lt - 4 + 9$, so $|1 - 2x|\lt5$.

Step2: Rewrite the absolute - value inequality as a compound inequality

If $|u|\lt a$ ($a\gt0$), then $-a\lt u\lt a$. Here $u = 1 - 2x$ and $a = 5$, so $-5\lt1 - 2x\lt5$.

Step3: Solve the left - hand side of the compound inequality

Solve $-5\lt1 - 2x$:
Subtract 1 from both sides: $-5-1\lt1 - 2x-1$, which gives $-6\lt - 2x$.
Divide both sides by $-2$ and reverse the inequality sign: $\frac{-6}{-2}\gt\frac{-2x}{-2}$, so $3\gt x$ or $x\lt3$.

Step4: Solve the right - hand side of the compound inequality

Solve $1 - 2x\lt5$:
Subtract 1 from both sides: $1 - 2x-1\lt5 - 1$, which gives $-2x\lt4$.
Divide both sides by $-2$ and reverse the inequality sign: $\frac{-2x}{-2}\gt\frac{4}{-2}$, so $x\gt - 2$.

The solution in interval notation is $(-2,3)$.

Answer:

A. The solution set in interval notation is $(-2,3)$; C. (The graph of the solution set $(-2,3)$ on the number - line has an open circle at $-2$, an open circle at 3, and a line segment between them)