Question

Write your own system of two linear inequalities.
Describe how you could determine whether a given point is a solution to your system of
linear inequalities. Include an example in your description.

Answer 1

### System of Linear Inequalities

Let's create the following system of linear inequalities:

1. \( 2x + 3y < 12 \)
2. \( x - y \geq 1 \)

### Determining if a Point is a Solution

To determine whether a given point \((x, y)\) is a solution to the system of linear inequalities, we will follow these steps:

1. **Substitute the coordinates** of the point into each inequality.
2. **Evaluate** each inequality to see if it is satisfied (i.e., true).
3. **Check for both inequalities**: The point is a solution to the system if it satisfies both inequalities.

### Example

Let's consider the point \((3, 2)\) and see if it is a solution to our system.

#### Step 1: Substitute the point into each inequality

**For the first inequality** \( 2x + 3y < 12 \):
- Substitute \(x = 3\) and \(y = 2\):
\[
2(3) + 3(2) < 12
\]
\[
6 + 6 < 12
\]
\[
12 < 12 \quad \text{(False)}
\]

**For the second inequality** \( x - y \geq 1 \):
- Substitute \(x = 3\) and \(y = 2\):
\[
3 - 2 \geq 1
\]
\[
1 \geq 1 \quad \text{(True)}
\]

#### Step 2: Evaluate the results
- The first inequality is **false**.
- The second inequality is **true**.

#### Step 3: Conclusion
The point \((3, 2)\) does not satisfy both inequalities; it only satisfies the second inequality. Therefore, \((3, 2)\) is **not a solution** to the system of inequalities.

### Additional Example

Let's try a different point: \((2, 2)\).

#### For the first inequality:
\[
2(2) + 3(2) < 12
\]
\[
4 + 6 < 12 \quad \text{(True)}
\]

#### For the second inequality:
\[
2 - 2 \geq 1
\]
\[
0 \geq 1 \quad \text{(False)}
\]

**Conclusion**: The point \((2, 2)\) is also **not a solution** to the system because it does not meet both inequalities.

To summarize, a point is a solution to the system of inequalities if the substituted values satisfy all inequalities simultaneously, which can be determined through direct substitution into each inequality.