Question
Solve the absolute inequality \( \begin{array}{l}|x+5|>3 \\ \text { a) }(-8,-2) \\ \text { b) }(-\infty,-8) \cup(-2, \infty) \\ \text { c) }(-2, \infty) \\ \text { d) }(-\infty,-8] \cup[-2, \infty)\end{array} \)
Ask by Simmons Bernard. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution to the inequality \( |x+5|>3 \) is \( x \in (-\infty,-8) \cup (-2, \infty) \).
Solution
Solve the equation \( |x+5|>3 \).
Solve the inequality by following steps:
- step0: Solve for \(x\):
\(\left|x+5\right|>3\)
- step1: Separate into possible cases:
\(\begin{align}&x+5>3\\&x+5<-3\end{align}\)
- step2: Solve the inequality:
\(\begin{align}&x>-2\\&x<-8\end{align}\)
- step3: Find the union:
\(x \in \left(-\infty,-8\right)\cup \left(-2,+\infty\right)\)
The solution to the absolute inequality \(|x+5|>3\) is \(x \in (-\infty,-8) \cup (-2, \infty)\).
Therefore, the correct answer is b) \((-8,-2) \cup (-\infty,-8) \cup (-2, \infty)\).
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Simplify this solution Mind Expander
To solve the absolute inequality \( |x+5| > 3 \), we start by rewriting it in two separate inequalities: 1. \( x + 5 > 3 \) 2. \( x + 5 < -3 \) Now, we solve each inequality: 1. For \( x + 5 > 3 \): \[ x > 3 - 5 \] \[ x > -2 \] 2. For \( x + 5 < -3 \): \[ x < -3 - 5 \] \[ x < -8 \] Combining the results gives us: \[ x < -8 \quad \text{or} \quad x > -2 \] In interval notation, the solution can be expressed as: \[ (-\infty, -8) \cup (-2, \infty) \] Thus, the correct answer is **b)** \((- \infty, -8) \cup (-2, \infty)\).
