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.