To solve the compound inequality \(-1 < \frac{4x - 2}{8} < 1\), we need to break it down into two separate inequalities and solve each one individually.

### Step 1: Separate the Compound Inequality
The compound inequality can be separated into:
1. \(-1 < \frac{4x - 2}{8}\)
2. \(\frac{4x - 2}{8} < 1\)

### Step 2: Solve Each Inequality

#### Inequality 1: \(-1 < \frac{4x - 2}{8}\)
1. Multiply all terms by 8 to clear the denominator:
   \[
   -1 \cdot 8 < 4x - 2 <=> -8 < 4x - 2
   \]
2. Add 2 to all terms to isolate the term with \(x\):
   \[
   -8 + 2 < 4x <=> -6 < 4x
   \]
3. Divide all terms by 4 to solve for \(x\):
   \[
   \frac{-6}{4} < x <=> -\frac{3}{2} < x
   \]

#### Inequality 2: \(\frac{4x - 2}{8} < 1\)
1. Multiply all terms by 8 to clear the denominator:
   \[
   \frac{4x - 2}{8} \cdot 8 < 1 \cdot 8 <=> 4x - 2 < 8
   \]
2. Add 2 to all terms to isolate the term with \(x\):
   \[
   4x - 2 + 2 < 8 + 2 <=> 4x < 10
   \]
3. Divide all terms by 4 to solve for \(x\):
   \[
   \frac{4x}{4} < \frac{10}{4} <=> x < \frac{5}{2}
   \]

### Step 3: Combine the Solutions
Combining the results from both inequalities, we get:
\[
-\frac{3}{2} < x < \frac{5}{2}
\]

### Final Answer
\[
-\frac{3}{2} < x < \frac{5}{2}
\]

Would you like more details or have any questions?

Here are some related questions to consider:

1. How do you solve compound inequalities involving absolute values?
2. What are the general steps to solve a system of linear inequalities?
3. How do you graph the solution set of a compound inequality on a number line?
4. What are some real-world applications of solving inequalities?
5. How can inequalities be used in optimization problems?

**Tip:** When solving compound inequalities, remember to keep track of the inequality direction when multiplying or dividing by negative numbers, as this will flip the inequality sign.

Learn how to solve the compound inequality -1 < (4x - 2)/8 < 1 through a detailed step-by-step explanation. Understand the key algebraic concepts and techniques, including isolating variables and handling inequalities. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation