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!