solve each inequality. Then graph the solution set on a number line. 3(9z+4)>35z-4

Let's solve the inequality step by step:

### Inequality:
\[
3(9z + 4) > 35z - 4
\]

### Step 1: Expand both sides
First, expand the left side of the inequality:

\[
3 \times 9z + 3 \times 4 > 35z - 4
\]
\[
27z + 12 > 35z - 4
\]

### Step 2: Move all terms involving \( z \) to one side
Subtract \( 27z \) from both sides:

\[
12 > 35z - 27z - 4
\]
\[
12 > 8z - 4
\]

### Step 3: Isolate \( z \)
Add 4 to both sides:

\[
12 + 4 > 8z
\]
\[
16 > 8z
\]

### Step 4: Solve for \( z \)
Divide both sides by 8:

\[
z < 2
\]

### Solution:
The solution to the inequality is:
\[
z < 2
\]

### Step 5: Graph the solution on a number line
To graph \( z < 2 \), draw a number line and mark an open circle at \( z = 2 \) (since \( 2 \) is not included in the solution), then shade the line to the left of 2 to represent all values less than 2.

---

Would you like me to graph this for you, or do you have any questions?

### 5 Relative Questions:
1. How do you solve inequalities with multiple terms on both sides?
2. What happens if you divide or multiply by a negative number in an inequality?
3. How do you represent solutions with "greater than or equal to" on a number line?
4. Can you have a solution where \( z \) is a range between two values?
5. What are the steps to check if a value is a solution to an inequality?

**Tip**: Always remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

Step-by-step solution for solving the linear inequality 3(9z + 4) > 35z - 4. Learn how to expand, simplify, and graph the solution set on a number line. This guide covers key algebraic concepts, including the distributive property and solving inequalities, suitable for students in Grades 8-10.

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