Solve each inequality. Then graph each answer, and write each answer in both set-builder
notation and interval notation. Show your work! (7 points each inequality) 5 + 3(2𝑥 − 3) ≥ 4𝑥 − 2

Let's solve the inequality step by step, graph the solution, and express the solution in both set-builder notation and interval notation.

### Step 1: Distribute and simplify the inequality
Given inequality:
\[
5 + 3(2x - 3) \geq 4x - 2
\]

Distribute the 3:
\[
5 + 6x - 9 \geq 4x - 2
\]

Simplify by combining like terms on the left side:
\[
6x - 4 \geq 4x - 2
\]

### Step 2: Move all terms involving \( x \) to one side
Subtract \( 4x \) from both sides:
\[
6x - 4x - 4 \geq -2
\]
\[
2x - 4 \geq -2
\]

### Step 3: Isolate the variable \( x \)
Add 4 to both sides:
\[
2x \geq 2
\]

Divide by 2:
\[
x \geq 1
\]

### Step 4: Express the solution in set-builder and interval notation
- **Set-builder notation:** \( \{ x \,|\, x \geq 1 \} \)
- **Interval notation:** \( [1, \infty) \)

### Step 5: Graph the solution
To graph the solution \( x \geq 1 \):

- Draw a number line.
- Place a solid dot at \( x = 1 \) (since \( x \) is greater than or equal to 1).
- Shade the region to the right of \( x = 1 \), extending towards positive infinity.

The graph shows that the solution includes 1 and all values greater than 1.

### Summary of the Answer
- **Set-builder notation:** \( \{ x \,|\, x \geq 1 \} \)
- **Interval notation:** \( [1, \infty) \)

### Graph:
1. Draw a number line.
2. Mark a solid dot at \( x = 1 \).
3. Shade the area to the right of \( x = 1 \).

---

Would you like to see a visual representation of the graph? Or do you have any other questions on this topic? 

Here are 8 related questions:

1. What happens if the inequality had been \( x > 1 \) instead of \( x \geq 1 \)?
2. How do you solve an inequality with a variable on both sides?
3. What does it mean to graph a strict inequality (e.g., \( x > a \))?
4. How do you interpret the solution of an inequality on a number line?
5. Can inequalities have solutions that include all real numbers?
6. How do you solve compound inequalities?
7. What is the difference between set-builder and interval notation?
8. How does multiplying or dividing by a negative number affect an inequality?

**Tip:** Always remember to flip the inequality sign when you multiply or divide both sides of an inequality by a negative number!

Learn how to solve and graph the inequality 5 + 3(2𝑥 − 3) ≥ 4𝑥 − 2 step-by-step. Understand set-builder and interval notation with detailed explanations and graphical representation. Suitable for students in Grades 7-9.

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