Write your own system of two linear inequalities.

Question

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.