System of Inequalities Analysis

The problem asks for the number of solutions to the following system of linear inequalities:

• $X_1 \ge 0$
• $X_2 \ge 0$
• $X_1+ X_2 \le 10$
• $2X_1+ 2X_2 \ge 22$

Simplifying the Inequalities

The first two inequalities define the first quadrant ($X_1$ and $X_2$ are non-negative).

The fourth inequality, $2X_1+ 2X_2 \ge 22$, can be simplified by dividing the entire inequality by 2:

$ \frac{2X_1+ 2X_2}{2} \ge \frac{22}{2} $

$ X_1+ X_2 \ge 11 $

The system effectively requires solutions satisfying:

• $X_1 \ge 0$
• $X_2 \ge 0$
• $X_1+ X_2 \le 10$
• $X_1+ X_2 \ge 11$

Identifying Contradictory Conditions

We need to find pairs $(X_1, X_2)$ that satisfy both $X_1+ X_2 \le 10$ and $X_1+ X_2 \ge 11$ simultaneously.

It is impossible for the sum of two numbers ($X_1+ X_2$) to be less than or equal to 10 and greater than or equal to 11 at the same time. These two conditions are mutually exclusive.

Conclusion on Number of Solutions

Because the conditions derived from the inequalities lead to a contradiction, no pair of values $(X_1, X_2)$ can satisfy all the inequalities at once.

Therefore, the number of solutions for this system of inequalities is 0.