https://file.aiquickdraw.com/mathbot/file/1728874163328sclhxroq.jpg

The inequality in the image is:

\[
\frac{7}{2}(5x - 15) + \frac{13}{2} > \frac{9x}{2}
\]

Let's solve this step by step.

### Step 1: Distribute the \(\frac{7}{2}\)
\[
\frac{7}{2} \times (5x - 15) = \frac{7}{2} \times 5x - \frac{7}{2} \times 15 = \frac{35x}{2} - \frac{105}{2}
\]
So the inequality becomes:
\[
\frac{35x}{2} - \frac{105}{2} + \frac{13}{2} > \frac{9x}{2}
\]

### Step 2: Combine like terms on the left side
\[
- \frac{105}{2} + \frac{13}{2} = -\frac{92}{2} = -46
\]
Thus, the inequality is now:
\[
\frac{35x}{2} - 46 > \frac{9x}{2}
\]

### Step 3: Eliminate the fractions by multiplying everything by 2
\[
35x - 92 > 9x
\]

### Step 4: Move \(9x\) to the left side
\[
35x - 9x > 92
\]
\[
26x > 92
\]

### Step 5: Solve for \(x\)
\[
x > \frac{92}{26}
\]
Simplifying the fraction:
\[
x > \frac{46}{13}
\]

### Conclusion:
The solution is \(x > \frac{46}{13}\), which is approximately \(x > 3.54\).

### Graphing on the number line:
On the number line, you would place an open circle at approximately \(3.54\) and shade to the right, indicating that \(x\) is greater than \(3.54\).

Would you like more details or have any questions? Here are 5 related questions that expand on this information:

1. What is the significance of using an open circle on the number line for this inequality?
2. How do you handle inequalities involving fractions in general?
3. How would the solution change if the inequality was \( \geq \) instead of \( > \)?
4. Can you explain why multiplying through by 2 helps to simplify this inequality?
5. How would the solution differ if there were negative coefficients involved in the inequality?

**Tip:** Always check your solution by substituting a number from the solution range into the original inequality to verify that it satisfies the inequality.

Solve the inequality 7/2(5x - 15) + 13/2 > 9x/2 and graph the solution on the number line.

Learn how to solve and graph the inequality 7/2(5x - 15) + 13/2 > 9x/2. This detailed step-by-step solution covers key algebraic concepts like distributing, combining like terms, and solving inequalities. Suitable for students in Grades 9-10.

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