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)

Explanation:

Step1: Simplify both sides

First, expand the right - hand side of the inequality.
For \(2(2 + x)\), using the distributive property \(a(b + c)=ab+ac\), we get \(2\times2+2\times x = 4 + 2x\).
For \(-2(1 + 2x)\), we get \(-2\times1-2\times2x=-2 - 4x\).
So the right - hand side of the inequality \(2(2 + x)-2(1 + 2x)=(4 + 2x)+(-2 - 4x)=4 + 2x-2 - 4x=2 - 2x\).
The original inequality \(9x-31<2(2 + x)-2(1 + 2x)\) becomes \(9x-31<2 - 2x\).

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

To get all the \(x\) terms on one side, we add \(2x\) to both sides of the inequality.
\(9x-31 + 2x<2-2x + 2x\)
Simplifying both sides, we have \(11x-31 < 2\).

Step3: Add 31 to both sides

To isolate the term with \(x\), we add 31 to both sides of the inequality.
\(11x-31 + 31<2 + 31\)
Simplifying both sides, we get \(11x<33\).

Step4: Divide both sides by 11

Divide both sides of the inequality \(11x<33\) by 11. Since 11 is a positive number, the direction of the inequality sign remains the same.
\(\frac{11x}{11}<\frac{33}{11}\)
Simplifying, we get \(x < 3\).

Answer:

\(x<3\)