Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Solving Absolute Value Inequalities
In the problem displayed, we are asked to solve the following inequality and express the answer using interval notation:
\[ |2x + 7| \ge 9 \]
To solve this inequality, follow these steps:
1. **Understand the Definition of Absolute Value Inequality**:
\[
|A| \ge B \quad \text{implies} \quad A \le -B \quad \text{or} \quad A \ge B
\]
2. **Apply the Definition**:
\[
|2x + 7| \ge 9 \quad \text{implies} \quad 2x + 7 \le -9 \quad \text{or} \quad 2x + 7 \ge 9
\]
3. **Solve Each Inequality Separately**:
For \(2x + 7 \le -9\):
\[
2x + 7 \le -9 \implies 2x \le -16 \implies x \le -8
\]
For \(2x + 7 \ge 9\):
\[
2x + 7 \ge 9 \implies 2x \ge 2 \implies x \ge 1
\]
4. **Combine Solutions**:
The solutions to these inequalities are \(x \le -8\) or \(x \ge 1\).
5. **Express in Interval Notation**:
\[
(-\infty, -8] \cup [1, \infty)
\]
There is no graph or diagram included in the problem. The solution can be visualized on a number line where the intervals are \( (-\infty, -8] \) and \([1, \infty) \).
Here is the solution graphically explained:
- For \( x \le -8 \), the interval includes all values from negative infinity up to and including \(-8\).
- For \( x \ge 1 \), the interval includes all values from \(1\) to positive infinity.
Thus, the complete solution in interval notation is:
\[
(-\infty, -8] \cup [1, \infty)
\]
This answer covers all possible values of \(x\) that satisfy](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ad8a3e2-d44f-406e-ad02-79162c0dd845%2Fbd265090-d9ad-4452-8813-826716da1167%2Ft4sc596_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Absolute Value Inequalities
In the problem displayed, we are asked to solve the following inequality and express the answer using interval notation:
\[ |2x + 7| \ge 9 \]
To solve this inequality, follow these steps:
1. **Understand the Definition of Absolute Value Inequality**:
\[
|A| \ge B \quad \text{implies} \quad A \le -B \quad \text{or} \quad A \ge B
\]
2. **Apply the Definition**:
\[
|2x + 7| \ge 9 \quad \text{implies} \quad 2x + 7 \le -9 \quad \text{or} \quad 2x + 7 \ge 9
\]
3. **Solve Each Inequality Separately**:
For \(2x + 7 \le -9\):
\[
2x + 7 \le -9 \implies 2x \le -16 \implies x \le -8
\]
For \(2x + 7 \ge 9\):
\[
2x + 7 \ge 9 \implies 2x \ge 2 \implies x \ge 1
\]
4. **Combine Solutions**:
The solutions to these inequalities are \(x \le -8\) or \(x \ge 1\).
5. **Express in Interval Notation**:
\[
(-\infty, -8] \cup [1, \infty)
\]
There is no graph or diagram included in the problem. The solution can be visualized on a number line where the intervals are \( (-\infty, -8] \) and \([1, \infty) \).
Here is the solution graphically explained:
- For \( x \le -8 \), the interval includes all values from negative infinity up to and including \(-8\).
- For \( x \ge 1 \), the interval includes all values from \(1\) to positive infinity.
Thus, the complete solution in interval notation is:
\[
(-\infty, -8] \cup [1, \infty)
\]
This answer covers all possible values of \(x\) that satisfy
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