Hampton Garza
12/17/2023 · Senior High School
Solve the absolute value inequality. \( -4|9-x|<-8 \) Select the correct answer below and, if necessary, fill in the answer box to complete your A. The solution set in interval notation is (Simplify your answer.) B. The solution set is \( \varnothing \).
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Step-by-step Solution
Solve the equation \( -4|9-x|<-8 \).
Solve the inequality by following steps:
- step0: Solve for \(x\):
\(-4\left|9-x\right|<-8\)
- step1: Change the sign:
\(4\left|9-x\right|>8\)
- step2: Divide both sides:
\(\frac{4\left|9-x\right|}{4}>\frac{8}{4}\)
- step3: Divide the numbers:
\(\left|9-x\right|>2\)
- step4: Separate into possible cases:
\(\begin{align}&9-x>2\\&9-x<-2\end{align}\)
- step5: Solve the inequality:
\(\begin{align}&x<7\\&x>11\end{align}\)
- step6: Find the union:
\(x \in \left(-\infty,7\right)\cup \left(11,+\infty\right)\)
The solution to the absolute value inequality \( -4|9-x|<-8 \) is \( x \in (-\infty,7) \cup (11,+\infty) \).
Therefore, the solution set in interval notation is \( (-\infty,7) \cup (11,+\infty) \).
Quick Answer
The solution set in interval notation is \( (-\infty,7) \cup (11,+\infty) \).
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