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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![### Math Inequalities
#### Problem Set
**(b)**
\[ \left| \frac{2x}{3} + \frac{4}{5} \right| - 1 \leq 2 \]
**(c)**
\[ \left| \frac{2x}{3} + \frac{4}{5} \right| - 1 \geq 2 \]
In this problem set, you are required to solve the inequalities involving absolute values. For each inequality:
1. Isolate the absolute value expression.
2. Split the inequality into two separate cases based on the definition of absolute value.
3. Solve each resulting inequality for the variable \( x \).
4. Combine the solutions to find the final solution set.
### Detailed Solution Steps for Problem (b):
For the inequality,
\[ \left| \frac{2x}{3} + \frac{4}{5} \right| - 1 \leq 2 \]
1. Add 1 to both sides to isolate the absolute value:
\[ \left| \frac{2x}{3} + \frac{4}{5} \right| \leq 3 \]
2. Write the inequality without the absolute value as two separate inequalities:
\[ -3 \leq \frac{2x}{3} + \frac{4}{5} \leq 3 \]
3. Solve the compound inequality for \( x \):
For \( -3 \leq \frac{2x}{3} + \frac{4}{5} \):
- Subtract \(\frac{4}{5}\) from both sides:
\[ -3 - \frac{4}{5} \leq \frac{2x}{3} \]
\[ -\frac{15}{5} - \frac{4}{5} \leq \frac{2x}{3} \]
\[ -\frac{19}{5} \leq \frac{2x}{3} \]
- Multiply both sides by 3:
\[ -\frac{57}{5} \leq 2x \]
- Divide both sides by 2:
\[ -\frac{57}{10} \leq x \]
For \( \frac{2x}{3} + \frac{4}{5} \le](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa19999fe-5ff0-4da5-8e70-0a6e9b11cafb%2F5af9f64b-79b1-4e65-b151-3947d3d76d5b%2Fcguilra_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Math Inequalities
#### Problem Set
**(b)**
\[ \left| \frac{2x}{3} + \frac{4}{5} \right| - 1 \leq 2 \]
**(c)**
\[ \left| \frac{2x}{3} + \frac{4}{5} \right| - 1 \geq 2 \]
In this problem set, you are required to solve the inequalities involving absolute values. For each inequality:
1. Isolate the absolute value expression.
2. Split the inequality into two separate cases based on the definition of absolute value.
3. Solve each resulting inequality for the variable \( x \).
4. Combine the solutions to find the final solution set.
### Detailed Solution Steps for Problem (b):
For the inequality,
\[ \left| \frac{2x}{3} + \frac{4}{5} \right| - 1 \leq 2 \]
1. Add 1 to both sides to isolate the absolute value:
\[ \left| \frac{2x}{3} + \frac{4}{5} \right| \leq 3 \]
2. Write the inequality without the absolute value as two separate inequalities:
\[ -3 \leq \frac{2x}{3} + \frac{4}{5} \leq 3 \]
3. Solve the compound inequality for \( x \):
For \( -3 \leq \frac{2x}{3} + \frac{4}{5} \):
- Subtract \(\frac{4}{5}\) from both sides:
\[ -3 - \frac{4}{5} \leq \frac{2x}{3} \]
\[ -\frac{15}{5} - \frac{4}{5} \leq \frac{2x}{3} \]
\[ -\frac{19}{5} \leq \frac{2x}{3} \]
- Multiply both sides by 3:
\[ -\frac{57}{5} \leq 2x \]
- Divide both sides by 2:
\[ -\frac{57}{10} \leq x \]
For \( \frac{2x}{3} + \frac{4}{5} \le
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