Question

solve the compound inequality for x. notice that this inequality contains a variable not only in the middle but also on the left and the right. when this occurs, solve by rewriting the inequality using the word and. for example, x < y < z can be rewritten as x < y and y < z.
9x - 31 < 2(2 + x) < - 2(1 + 2x)
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is \\((-\infty, 3)\\).
(type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the expression.)
b. the solution set is \\(\varnothing\\).

Explanation:

Step1: Expand both sides

First, expand the right - hand side of the inequality \(9x - 31<2(2 + x)-2(1 + 2x)\).
Using the distributive property \(a(b + c)=ab+ac\), we have:
\(2(2 + x)=4 + 2x\) and \(-2(1 + 2x)=-2-4x\).
So the right - hand side becomes \(4 + 2x-2 - 4x=(4 - 2)+(2x-4x)=2-2x\).
The inequality is now \(9x-31 < 2-2x\).

Step2: Add \(2x\) to both sides

Add \(2x\) to both sides of the inequality \(9x-31 < 2-2x\) to get:
\(9x+2x-31<2-2x + 2x\)
\(11x-31 < 2\).

Step3: Add 31 to both sides

Add 31 to both sides of the inequality \(11x-31 < 2\):
\(11x-31 + 31<2 + 31\)
\(11x<33\).

Step4: Divide both sides by 11

Divide both sides of the inequality \(11x<33\) by 11 (since \(11>0\), the direction of the inequality sign remains the same):
\(\frac{11x}{11}<\frac{33}{11}\)
\(x < 3\).

In interval notation, the solution set of \(x < 3\) is \((-\infty,3)\).

Answer:

\((-\infty,3)\)