Let's solve the compound inequality \( 4x - 1 > -9 \) or \( 7x - 2 \leq 19 \) step by step.

### First Inequality: \( 4x - 1 > -9 \)
1. Add 1 to both sides to isolate the term with \( x \):
   \[
   4x > -8
   \]
2. Divide both sides by 4 to solve for \( x \):
   \[
   x > -2
   \]

### Second Inequality: \( 7x - 2 \leq 19 \)
1. Add 2 to both sides:
   \[
   7x \leq 21
   \]
2. Divide both sides by 7 to solve for \( x \):
   \[
   x \leq 3
   \]

### Combined Solution
Since the inequalities are connected by "or," the solution set includes values of \( x \) that satisfy either inequality. 

Thus, the combined solution is:

\[
x > -2 \quad \text{or} \quad x \leq 3
\]

### Interpretation
This means any \( x \) greater than \( -2 \) or any \( x \) less than or equal to \( 3 \) satisfies the compound inequality.

### Final Answer:
Since any \( x > -2 \) already covers all \( x \leq 3 \) as well, the complete solution is \( x \in (-2, \infty) \), which simplifies to all real numbers. However, for clarity:

\[
\text{The solution is } x \in (-2, \infty) \quad \text{since all numbers greater than } -2 \text{ are included.}
\]

---

Would you like further clarification or have any additional questions?

Here are some related questions:
1. How do you solve compound inequalities with "and" instead of "or"?
2. How do you graph \( x > -2 \) on a number line?
3. What are the possible cases where a compound inequality might have no solution?
4. How does the solution change if the inequalities involve strict inequalities (e.g., \( < \) instead of \( \leq \))?
5. Can you provide an example of a word problem that involves solving compound inequalities?

**Tip:** Always carefully analyze the logical connector ("and" vs. "or") when solving compound inequalities. This determines whether the solutions overlap (for "and") or are combined (for "or").

Learn how to solve compound inequalities 4x - 1 > -9 or 7x - 2 <= 19 with a detailed step-by-step solution. Understand the interpretation and combined solution of the compound inequality involving both 'or' conditions. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation