Question
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Tutor-Verified Answer
Answer
\(m \in (-\infty,6] \cup [10,+\infty)\)
Solution
Solve the equation \( 2|-8+m| \geq 4 \).
Solve the inequality by following steps:
- step0: Solve for \(m\):
\(2\left|-8+m\right|\geq 4\)
- step1: Calculate the absolute value:
\(2\left|8-m\right|\geq 4\)
- step2: Divide both sides:
\(\frac{2\left|8-m\right|}{2}\geq \frac{4}{2}\)
- step3: Divide the numbers:
\(\left|8-m\right|\geq 2\)
- step4: Separate into possible cases:
\(\begin{align}&8-m\geq 2\\&8-m\leq -2\end{align}\)
- step5: Solve the inequality:
\(\begin{align}&m\leq 6\\&m\geq 10\end{align}\)
- step6: Find the union:
\(m \in \left(-\infty,6\right]\cup \left[10,+\infty\right)\)
The solution to the inequality \(2|-8+m| \geq 4\) is \(m \in (-\infty,6] \cup [10,+\infty)\).
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Simplify this solution The Deep Dive
To solve the inequality \( 2|-8+m| \geq 4 \), we first simplify it by dividing both sides by 2, giving us \( |-8+m| \geq 2 \). This absolute value inequality leads to two cases: \( -8+m \geq 2 \) and \( -8+m \leq -2 \). For the first case, solving \( -8+m \geq 2 \) gives \( m \geq 10 \). For the second case, solving \( -8+m \leq -2 \) yields \( m \leq 6 \). Therefore, the solution to the inequality is \( m \leq 6 \) or \( m \geq 10 \).
