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 Compound Inequalities
In this exercise, you are given a compound inequality and need to solve for the variable \( x \).
#### Problem
Solve for \( x \):
\[ -5 \leq 10x + 13 \leq 7 \]
**Instructions**: Write your answer in interval notation in the provided box.
#### Steps to Solve:
1. **Isolate \( x \)**:
- The compound inequality can be split into two separate inequalities:
\[ -5 \leq 10x + 13 \]
\[ 10x + 13 \leq 7 \]
2. **Solve each inequality separately**:
- For the first inequality:
\[ -5 \leq 10x + 13 \]
Subtract 13 from both sides:
\[ -5 - 13 \leq 10x \]
\[ -18 \leq 10x \]
Divide by 10:
\[ -\frac{18}{10} \leq x \]
Simplify the fraction:
\[ -1.8 \leq x \]
- For the second inequality:
\[ 10x + 13 \leq 7 \]
Subtract 13 from both sides:
\[ 10x \leq 7 - 13 \]
\[ 10x \leq -6 \]
Divide by 10:
\[ x \leq -\frac{6}{10} \]
Simplify the fraction:
\[ x \leq -0.6 \]
3. **Combine the solutions**:
- The solutions of the inequalities, \( -1.8 \leq x \) and \( x \leq -0.6 \), together form:
\[ -1.8 \leq x \leq -0.6 \]
#### Solution in Interval Notation
\[ [-1.8, -0.6] \]
Use the provided box to write your answer in the form of an interval.
Box:
\[ \boxed{[-1.8, -0.6]} \]
This detailed step-by-step solution helps the learners to understand the process of solving compound inequalities and writing the solution in interval notation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ed3be98-469b-4630-9026-6d8c89f34a56%2Fc09e2d76-f237-414a-bf66-9d4ff4cc14f6%2Fvsc75o7_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Compound Inequalities
In this exercise, you are given a compound inequality and need to solve for the variable \( x \).
#### Problem
Solve for \( x \):
\[ -5 \leq 10x + 13 \leq 7 \]
**Instructions**: Write your answer in interval notation in the provided box.
#### Steps to Solve:
1. **Isolate \( x \)**:
- The compound inequality can be split into two separate inequalities:
\[ -5 \leq 10x + 13 \]
\[ 10x + 13 \leq 7 \]
2. **Solve each inequality separately**:
- For the first inequality:
\[ -5 \leq 10x + 13 \]
Subtract 13 from both sides:
\[ -5 - 13 \leq 10x \]
\[ -18 \leq 10x \]
Divide by 10:
\[ -\frac{18}{10} \leq x \]
Simplify the fraction:
\[ -1.8 \leq x \]
- For the second inequality:
\[ 10x + 13 \leq 7 \]
Subtract 13 from both sides:
\[ 10x \leq 7 - 13 \]
\[ 10x \leq -6 \]
Divide by 10:
\[ x \leq -\frac{6}{10} \]
Simplify the fraction:
\[ x \leq -0.6 \]
3. **Combine the solutions**:
- The solutions of the inequalities, \( -1.8 \leq x \) and \( x \leq -0.6 \), together form:
\[ -1.8 \leq x \leq -0.6 \]
#### Solution in Interval Notation
\[ [-1.8, -0.6] \]
Use the provided box to write your answer in the form of an interval.
Box:
\[ \boxed{[-1.8, -0.6]} \]
This detailed step-by-step solution helps the learners to understand the process of solving compound inequalities and writing the solution in interval notation.
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