Question

question
solve the following inequality algebraically.
$2|3 + 3x| - 1 > 23$

Explanation:

Step1: Isolate the absolute value term

Add 1 to both sides, then divide by 2.

$$\begin{align*} 2|3+3x|-1+1&>23+1\\ 2|3+3x|&>24\\ \frac{2|3+3x|}{2}&>\frac{24}{2}\\ |3+3x|&>12 \end{align*}$$

Step2: Split into two inequalities

For $|A|>B$, solve $A>B$ and $A<-B$.

$$ 3+3x>12 \quad \text{or} \quad 3+3x<-12 $$

Step3: Solve first inequality

Subtract 3, then divide by 3.

$$\begin{align*} 3+3x-3&>12-3\\ 3x&>9\\ \frac{3x}{3}&>\frac{9}{3}\\ x&>3 \end{align*}$$

Step4: Solve second inequality

Subtract 3, then divide by 3.

$$\begin{align*} 3+3x-3&<-12-3\\ 3x&<-15\\ \frac{3x}{3}&<\frac{-15}{3}\\ x&<-5 \end{align*}$$

Answer:

$x < -5$ or $x > 3$