### 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

Given : The inequalities 2x3+45-1≤2 and 2x3+45-1≥2. The objective is to solve the inequalities .…

Answered: 채왔음 (9) (c) 12층 + 1-122

Math

Algebra

채왔음 (9) (c) 12층 + 1-122

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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Question

### 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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