Hi Math.Bot! I am helping my child with their math homework. Today, my child’s class is having a discussion. The discussion topic is 1-Variable Equations and Inequalities. Is it possible for you to assist me with this discussion?

Of course! I'd be happy to assist with a discussion on **1-Variable Equations and Inequalities**. Let's break it down into key points:

### 1-Variable Equations:
A **1-variable equation** is an equation where there is only one unknown (variable), usually represented by a letter like \( x \). The goal is to solve for the value of the variable that makes the equation true.

**Example:**
\[
2x + 5 = 9
\]
To solve for \( x \):
\[
2x = 9 - 5 \quad \text{(subtract 5 from both sides)}
\]
\[
2x = 4
\]
\[
x = \frac{4}{2} = 2 \quad \text{(divide by 2)}
\]
So, \( x = 2 \) is the solution.

### 1-Variable Inequalities:
A **1-variable inequality** involves an inequality symbol (\( <, \leq, >, \geq \)) instead of an equals sign.

**Example:**
\[
3x - 4 \leq 8
\]
To solve for \( x \):
\[
3x \leq 8 + 4 \quad \text{(add 4 to both sides)}
\]
\[
3x \leq 12
\]
\[
x \leq \frac{12}{3} = 4 \quad \text{(divide by 3)}
\]
So, \( x \leq 4 \) is the solution, meaning \( x \) can be any value less than or equal to 4.

### Key Concepts for the Discussion:
1. **Balancing the equation**: When solving, the goal is to isolate the variable. You perform the same operation on both sides to maintain balance.
2. **Types of solutions**: Equations usually have one solution, but inequalities have a range of solutions.
3. **Graphing inequalities**: Inequalities can be represented on a number line, where:
   - \( < \) or \( > \) is shown with an open circle (not included).
   - \( \leq \) or \( \geq \) is shown with a closed circle (included).
4. **Inverse operations**: Use inverse operations (e.g., subtraction vs. addition) to isolate the variable.
5. **Checking solutions**: Substitute the solution back into the original equation or inequality to verify correctness.

### Sample Problem for Discussion:
Solve the inequality \( 5x - 7 > 18 \).

**Solution:**
\[
5x > 18 + 7
\]
\[
5x > 25
\]
\[
x > \frac{25}{5} = 5
\]
So, \( x > 5 \) is the solution.

Would you like more details on any part, or do you have any specific questions about equations or inequalities?

### Related Questions:
1. How do you graph the solution to an inequality like \( x \geq 3 \)?
2. What happens if you multiply or divide both sides of an inequality by a negative number?
3. Can you have equations that have no solution or infinite solutions?
4. How do you handle absolute value inequalities?
5. What are some real-life examples of one-variable equations and inequalities?

### Tip:
When solving inequalities, remember that multiplying or dividing both sides by a negative number *flips* the inequality sign!

Explore the concepts of 1-variable equations and inequalities, including balancing equations, solving step-by-step, and graphing inequalities. This guide is perfect for students in Grades 6-8, covering essential algebra topics with examples and explanations.

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