Solve the inequality. Graph the solution.
negative 6​(6negative 3​x)greater than36
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Part 1
The solution is 
  
enter your response here.

Let's solve the inequality step by step.

### Given Inequality:
\[
-6 \left( 6 - \frac{3}{x} \right) > 36
\]

### Step 1: Distribute the \(-6\) inside the parentheses.
\[
-6 \cdot 6 + (-6) \cdot \left(-\frac{3}{x}\right) > 36
\]
\[
-36 + \frac{18}{x} > 36
\]

### Step 2: Add 36 to both sides to isolate the fraction term.
\[
\frac{18}{x} > 72
\]

### Step 3: Multiply both sides by \(x\) to eliminate the fraction.
Note: Since we are multiplying by \(x\), we need to consider the sign of \(x\). The inequality will flip if \(x\) is negative.
1. If \(x > 0\):
\[
18 > 72x
\]
\[
x < \frac{18}{72} = \frac{1}{4}
\]
2. If \(x < 0\):
\[
18 < 72x
\]
\[
x > \frac{1}{4}
\]

### Step 4: Combine the results.
Since \(x\) cannot be both greater and less than \(\frac{1}{4}\) at the same time, the inequality has no solution for \(x > 0\). We only consider \(x < 0\), which leads to \(x > \frac{1}{4}\) under the assumption that \(x < 0\).

However, this is a contradiction because \(\frac{1}{4}\) is positive. Therefore, the only valid solution occurs for \(x < 0\).

### Solution:
The solution is \(x < 0\).

### Graphing the Solution:
On a number line, shade the region to the left of 0, indicating that all values of \(x\) less than 0 satisfy the inequality.

Would you like to see any specific details, or do you have any questions? Here are some related questions:

1. What happens if the inequality involves a quadratic expression?
2. How would the solution change if the inequality was non-strict (i.e., \( \geq \) instead of \( > \))?
3. How do you solve an inequality involving absolute values?
4. Can you explain how to graph inequalities on a coordinate plane?
5. How does multiplying or dividing by a negative number affect the direction of the inequality?

**Tip:** Always check the sign of \(x\) when solving inequalities that involve multiplying or dividing by variables to determine if the inequality direction will flip.

Solve the inequality. Graph the solution. negative 6​(6 - negative 3​x) greater than 36

Explore the solution to the linear inequality -6(6 - 3/x) > 36 with a step-by-step approach. This problem demonstrates key algebraic concepts, including distribution, properties of inequalities, and graphing solutions. Suitable for students in Grades 9-10.

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