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

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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} (4\mleft|x+9\mright|)/(4)>(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}

User Samer Buna
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