To determine which ordered pair is in the solution set of the given system of linear inequalities, we need to carefully analyze and test each point against both inequalities. The inequalities are:
[tex]\[
\begin{array}{l}
y > \frac{3}{2}x - 1 \\
y < \frac{3}{2}x - 1
\end{array}
\][/tex]
Let's examine each inequality one by one.
### Inequality 1:
[tex]\[ y > \frac{3}{2}x - 1 \][/tex]
### Inequality 2:
[tex]\[ y < \frac{3}{2}x - 1 \][/tex]
Both inequalities involve the same line represented by [tex]\( y = \frac{3}{2}x - 1 \)[/tex].
### Step-by-Step Analysis:
#### For the ordered pair (-5, 2):
1. Calculate [tex]\(\frac{3}{2}(-5) - 1\)[/tex]:
[tex]\[
\frac{3}{2} \cdot (-5) - 1 = -\frac{15}{2} - 1 = -7.5 - 1 = -8.5
\][/tex]
2. Check if [tex]\(2 > -8.5\)[/tex]: This is true.
3. Check if [tex]\(2 < -8.5\)[/tex]: This is false.
Thus, (-5, 2) does not satisfy both inequalities.
#### For the ordered pair (2, 2):
1. Calculate [tex]\(\frac{3}{2}(2) - 1\)[/tex]:
[tex]\[
\frac{3}{2} \cdot 2 - 1 = 3 - 1 = 2
\][/tex]
2. Check if [tex]\(2 > 2\)[/tex]: This is false.
3. Check if [tex]\(2 < 2\)[/tex]: This is also false.
Thus, (2, 2) does not satisfy both inequalities.
#### For the ordered pair (5, 2):
1. Calculate [tex]\(\frac{3}{2}(5) - 1\)[/tex]:
[tex]\[
\frac{3}{2} \cdot 5 - 1 = \frac{15}{2} - 1 = 7.5 - 1 = 6.5
\][/tex]
2. Check if [tex]\(2 > 6.5\)[/tex]: This is false.
3. Check if [tex]\(2 < 6.5\)[/tex]: This is true.
Thus, (5, 2) does not satisfy both inequalities.
### Conclusion:
Given that we need the ordered pairs to satisfy both inequalities simultaneously, we see that none of the given points [tex]\((-5, 2)\)[/tex], [tex]\((2, 2)\)[/tex], and [tex]\((5, 2)\)[/tex] satisfy both inequalities. Therefore, the solution set is empty.
Hence, there are no ordered pairs in the solution set of the given system of linear inequalities.
