Let's break down the system of inequalities step by step and figure out how to represent $S$ as a union of two sets based on the variable $x$.

The system is:

$y \geq \frac{10}{9}x + \frac{105}{9}$ $y \leq 21x + 131$ $y < -\frac{11}{8}x + \frac{153}{8}$

Step 1: Finding the intersection points

To determine in what intervals $x$ varies, we need to find the points where the boundary lines intersect. This gives us the critical values for $x$.

Intersection of $y = \frac{10}{9}x + \frac{105}{9}$ and $y = 21x + 131$

Equating both expressions for $y$:

$\frac{10}{9}x + \frac{105}{9} = 21x + 131$

Multiply everything by 9 to clear the denominators:

$10x + 105 = 189x + 1179$

Simplifying:

$10x - 189x = 1179 - 105$ $-179x = 1074$ $x = \frac{1074}{179} = 6$

So, the first intersection occurs at $x = 6$.

Intersection of $y = 21x + 131$ and $y = -\frac{11}{8}x + \frac{153}{8}$

Equating both expressions for $y$:

$21x + 131 = -\frac{11}{8}x + \frac{153}{8}$

Multiply through by 8 to eliminate the fractions:

$168x + 1048 = -11x + 153$

Simplifying:

$168x + 11x = 153 - 1048$ $179x = -895$ $x = \frac{-895}{179} = -5$

So, the second intersection occurs at $x = -5$.

Step 2: Determining the intervals

Now that we have the critical points $x = -5$ and $x = 6$, we can describe the intervals for $x$:

1. For $x \in (-\infty, -5]$, $S$ will be the region defined by the inequalities involving the lines $y \geq \frac{10}{9}x + \frac{105}{9}$ and $y < -\frac{11}{8}x + \frac{153}{8}$.
2. For $x \in [-5, 6]$, $S$ will be the region bounded by $y \geq \frac{10}{9}x + \frac{105}{9}$ and $y \leq 21x + 131$.
3. For $x \in [6, \infty)$, $S$ will be the region between $y \leq 21x + 131$ and $y < -\frac{11}{8}x + \frac{153}{8}$.

Thus, the feasible set $S$ is a union of these intervals.

Final Answer:

The intervals where $x$ varies are:

• $(-\infty, -5]$
• $[-5, 6]$
• $[6, \infty)$

Would you like more detailed explanations or have any other questions?

Five Related Questions:

1. How do you graphically represent the feasible region for this system of inequalities?
2. What happens to the system if we change the inequality symbols in one of the equations?
3. How would the solution change if we added another inequality constraint to the system?
4. What is the geometric meaning of the intersection points we found?
5. Can this system be solved using matrix methods or linear programming?

Tip:

Always check the slope and intercept of each line to understand how the inequalities divide the plane. This helps visualize the solution.