Pregunta
Solve: \( |2 x-3| \geq 4 \) Give your answer as an interval using whole numbers, proper fractions or improper fractions. ?
Ask by Osborne Ryan. in the United States
Jan 24,2025
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The solution is \( x \leq -\frac{1}{2} \) or \( x \geq \frac{7}{2} \).
Solución
Solve the equation \( |2x-3| \geq 4 \).
Solve the inequality by following steps:
- step0: Solve for \(x\):
\(\left|2x-3\right|\geq 4\)
- step1: Separate into possible cases:
\(\begin{align}&2x-3\geq 4\\&2x-3\leq -4\end{align}\)
- step2: Solve the inequality:
\(\begin{align}&x\geq \frac{7}{2}\\&x\leq -\frac{1}{2}\end{align}\)
- step3: Find the union:
\(x \in \left(-\infty,-\frac{1}{2}\right]\cup \left[\frac{7}{2},+\infty\right)\)
The solution to the inequality \(|2x-3| \geq 4\) is \(x \in (-\infty, -\frac{1}{2}) \cup (\frac{7}{2}, +\infty)\).
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Extra Insights
To solve the inequality \( |2x - 3| \geq 4 \), we break it into two cases: 1. \( 2x - 3 \geq 4 \): \[ 2x \geq 7 \quad \Rightarrow \quad x \geq \frac{7}{2} \] 2. \( 2x - 3 \leq -4 \): \[ 2x \leq -1 \quad \Rightarrow \quad x \leq -\frac{1}{2} \] Now, combining both cases, we find the solution is: \[ x \leq -\frac{1}{2} \quad \text{or} \quad x \geq \frac{7}{2} \] In interval notation, the answer is: \[ (-\infty, -\frac{1}{2}] \cup [\frac{7}{2}, \infty) \]
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