Question

Solve the following inequality algebraically:4|x+9|-2>10

Answer 1

The Solution:

Given the inequality below:

$4\mleft|x+9\mright|-2>10$

We are required to solve the above inequality.

$\begin{gathered} 4\mleft|x+9\mright|-2>10 \\ \text{ Add 2 to both sides, we get} \\ 4\mleft|x+9\mright|-2+2>10+2 \\ 4\mleft|x+9\mright|>12 \end{gathered}$

Dividing both sides by 4, we get

$\begin{gathered} \frac{4\mleft|x+9\mright|}{4}>\frac{12}{4} \\ \\ \mleft|x+9\mright|>3 \end{gathered}$

Applying the absolute rule that states that:

$\begin{gathered} \mleft|x\mright|>a,a>0\text{ means} \\ x<-a\text{ or } \\ x>a \end{gathered}$

We have:

$\begin{gathered} x+9<-3\text{ or} \\ x+9>3 \end{gathered}$

Solving each of them, we have

$\begin{gathered} x+9<-3 \\ x<-3-9 \\ x<-12 \end{gathered}$

Or

$\begin{gathered} x+9>3 \\ x>3-9 \\ x>-6 \end{gathered}$

Therefore, the correct answer is:

$\begin{gathered} x<-12\text{ or} \\ x>-6 \end{gathered}$