Solve and write the solution to the inequality in both interval and set notation. 3(2x + 7)/-3 > x + 1

Let's solve the inequality:

\[
\frac{3(2x + 7)}{-3} > x + 1
\]

### Step 1: Simplify the left-hand side of the inequality.
The \(3\) in the numerator and the \(-3\) in the denominator cancel out, leaving:

\[
-(2x + 7) > x + 1
\]

### Step 2: Distribute the negative sign on the left-hand side.
\[
-2x - 7 > x + 1
\]

### Step 3: Move all terms involving \(x\) to one side.
Add \(2x\) to both sides:
\[
-7 > 3x + 1
\]

### Step 4: Isolate \(x\).
Subtract 1 from both sides:
\[
-7 - 1 > 3x
\]
\[
-8 > 3x
\]

### Step 5: Solve for \(x\).
Divide both sides by 3:
\[
\frac{-8}{3} > x
\]
or equivalently,
\[
x < \frac{-8}{3}
\]

### Step 6: Express the solution in interval and set notation.

- **Interval notation**: \(\left(-\infty, \frac{-8}{3}\right)\)
- **Set notation**: \(\{x \in \mathbb{R} \mid x < \frac{-8}{3}\}\)

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Would you like more details or have any questions?

### Relative Questions:
1. How do you handle inequalities when multiplying or dividing by negative numbers?
2. What are the key differences between solving equations and inequalities?
3. How do you convert between set and interval notation?
4. What would the solution look like if the inequality were \( \geq \) instead of \( > \)?
5. What is the graphical representation of the solution to this inequality?

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

Solve the inequality 3(2x + 7)/-3 > x + 1 with a detailed step-by-step solution. This problem involves simplifying algebraic expressions and solving linear inequalities. Learn how to express the solution in both interval and set notation. Suitable for students in Grades 8-10.

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