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D. If the original inequality of interest were instead 0 < |x − 2| < 3, exactly one real number must be excluded from your solution interval (as compared to your solution interval from part A). Which real number is excluded in this case? Explain. E. Effectively, your result from part D implies that the inequality 0 < x − 2| < 3 has a pair of solution intervals, each of width 3. Explain.

D. If the original inequality of interest were instead 0 < |x − 2| < 3, exactly one real number must be excluded from your solution interval (as compared to your solution interval from part A). Which real number is excluded in this case? Explain. E. Effectively, your result from part D implies that the inequality 0 < x − 2| < 3 has a pair of solution intervals, each of width 3. Explain.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I only need help with solutions to questions D and E. I do not need solutions to A,B, and C. Thank you so much!
Below is a transcription and explanation of the educational text on solving inequalities. --- **4. Consider the inequality \(|x - 2| < 3\). As needed, reference back to the terminology of Exercise 3 above.** **A.** State the solution to this inequality in interval notation. **B.** Consider your solution interval from part A. What is the width of the resulting solution interval? What is its midpoint? What is its radius? **C.** If the original inequality of interest were instead \(|x - 2| \leq 3\), exactly two additional real numbers must be included in your solution interval (as compared to your solution interval from part A). Which two real numbers are included in this case? Explain. **D.** If the original inequality of interest were instead \(0 < |x - 2| < 3\), exactly one real number must be excluded from your solution interval (as compared to your solution interval from part A). Which real number is excluded in this case? Explain. **E.** Effectively, your result from part D implies that the inequality \(0 < |x - 2| < 3\) has a pair of solution intervals, each of width 3. Explain. --- **Explanation:** - **Part A:** This exercise involves expressing the solution of an absolute value inequality \(|x - 2| < 3\) in interval notation. - **Part B:** This part asks for analysis of the solution interval: its width, midpoint, and radius. - **Part C:** Modifying the inequality to \(|x - 2| \leq 3\) requires identifying additional numbers included in the interval. - **Part D:** With the inequality \(0 < |x - 2| < 3\), the task is to determine which number must be excluded from the solution interval. - **Part E:** The exercise connects the results from Part D to identify two separate intervals created by the modified inequality. This sequence of exercises aids in the understanding of absolute value inequalities and their representation on the number line.
Transcribed Image Text:Below is a transcription and explanation of the educational text on solving inequalities. --- **4. Consider the inequality \(|x - 2| < 3\). As needed, reference back to the terminology of Exercise 3 above.** **A.** State the solution to this inequality in interval notation. **B.** Consider your solution interval from part A. What is the width of the resulting solution interval? What is its midpoint? What is its radius? **C.** If the original inequality of interest were instead \(|x - 2| \leq 3\), exactly two additional real numbers must be included in your solution interval (as compared to your solution interval from part A). Which two real numbers are included in this case? Explain. **D.** If the original inequality of interest were instead \(0 < |x - 2| < 3\), exactly one real number must be excluded from your solution interval (as compared to your solution interval from part A). Which real number is excluded in this case? Explain. **E.** Effectively, your result from part D implies that the inequality \(0 < |x - 2| < 3\) has a pair of solution intervals, each of width 3. Explain. --- **Explanation:** - **Part A:** This exercise involves expressing the solution of an absolute value inequality \(|x - 2| < 3\) in interval notation. - **Part B:** This part asks for analysis of the solution interval: its width, midpoint, and radius. - **Part C:** Modifying the inequality to \(|x - 2| \leq 3\) requires identifying additional numbers included in the interval. - **Part D:** With the inequality \(0 < |x - 2| < 3\), the task is to determine which number must be excluded from the solution interval. - **Part E:** The exercise connects the results from Part D to identify two separate intervals created by the modified inequality. This sequence of exercises aids in the understanding of absolute value inequalities and their representation on the number line.
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