X+1/3 + x-3/5  >2  a. Rewrite the inequality.   C. Construct a table of signs Interval  Test Point  b. Plot the points on the number line.  D. Summarize the interval

Let's break down the problem into its parts to solve it systematically.

### **a. Rewrite the Inequality**

Given inequality: 

\[
\frac{X + 1}{3} + \frac{X - 3}{5} > 2
\]

The first step is to eliminate the fractions by finding a common denominator. The least common denominator (LCD) of 3 and 5 is 15. Multiply every term by 15:

\[
15 \times \left(\frac{X + 1}{3} + \frac{X - 3}{5}\right) > 15 \times 2
\]

This simplifies to:

\[
5(X + 1) + 3(X - 3) > 30
\]

Expanding the terms:

\[
5X + 5 + 3X - 9 > 30
\]

Combine like terms:

\[
8X - 4 > 30
\]

Add 4 to both sides:

\[
8X > 34
\]

Finally, divide by 8:

\[
X > \frac{34}{8} = \frac{17}{4} = 4.25
\]

### **b. Plot the Points on the Number Line**

Since \( X > 4.25 \), you would plot an open circle at \( X = 4.25 \) on the number line and shade to the right, indicating that \( X \) can take any value greater than 4.25.

### **c. Construct a Table of Signs**

To construct a table of signs, consider the inequality \( X - 4.25 > 0 \) (since we have \( X > 4.25 \)).

| Interval        | Test Point | Sign of Expression \( X - 4.25 \) |
|-----------------|------------|-----------------------------------|
| \( (-\infty, 4.25) \) | 0          | Negative (-)                      |
| \( (4.25, \infty) \)  | 5          | Positive (+)                      |

### **d. Summarize the Interval**

The solution to the inequality is the interval:

\[
X \in (4.25, \infty)
\]

This means that \( X \) can take any value greater than 4.25.

### Summary
- The inequality simplifies to \( X > 4.25 \).
- The number line would show an open circle at 4.25, with shading to the right.
- The table of signs indicates the expression is positive when \( X > 4.25 \).
- The interval solution is \( (4.25, \infty) \).

Would you like more details on any part of this solution, or do you have any questions?

### 5 Related Questions:
1. How do you solve inequalities involving absolute values?
2. What are some techniques to solve quadratic inequalities?
3. Can you explain how to solve inequalities with multiple variables?
4. How do you graph inequalities in two variables?
5. What is the difference between strict and non-strict inequalities?

### Tip:
Always check your solution by substituting a value from the solution interval back into the original inequality.

Learn how to solve a complex inequality involving fractions step-by-step. This problem covers key algebraic concepts such as inequalities, fraction operations, and number line representation. Suitable for students in Grades 9-12.

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