Solve the inequality. Express the answer using interval notation. |2x + 7| 2 9

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,...
Question
### 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
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
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Knowledge Booster
Inequality
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education