To determine which points lie in the solution set of the given system of inequalities:
[tex]\[
\begin{array}{l}
y > -3x + 3 \\
y > x + 2
\end{array}
\][/tex]
we will examine each point against both inequalities.
The points to be tested are:
1. [tex]\((2, -5)\)[/tex]
2. [tex]\((-2, 5)\)[/tex]
3. [tex]\((2, 5)\)[/tex]
4. [tex]\((-2, -5)\)[/tex]
Let's test each point step-by-step:
### Point [tex]\((2, -5)\)[/tex]
- For [tex]\(y > -3x + 3\)[/tex]:
[tex]\( -5 > -3(2) + 3 \)[/tex]
[tex]\( -5 > -6 + 3 \)[/tex]
[tex]\( -5 > -3 \)[/tex] (False)
Since one of the inequalities is not satisfied, [tex]\((2, -5)\)[/tex] does not lie in the solution set.
### Point [tex]\((-2, 5)\)[/tex]
- For [tex]\(y > -3x + 3\)[/tex]:
[tex]\( 5 > -3(-2) + 3 \)[/tex]
[tex]\( 5 > 6 + 3 \)[/tex]
[tex]\( 5 > 9 \)[/tex] (False)
Since one of the inequalities is not satisfied, [tex]\((-2, 5)\)[/tex] does not lie in the solution set.
### Point [tex]\((2, 5)\)[/tex]
- For [tex]\(y > -3x + 3\)[/tex]:
[tex]\( 5 > -3(2) + 3 \)[/tex]
[tex]\( 5 > -6 + 3 \)[/tex]
[tex]\( 5 > -3 \)[/tex] (True)
- For [tex]\(y > x + 2\)[/tex]:
[tex]\( 5 > 2 + 2 \)[/tex]
[tex]\( 5 > 4 \)[/tex] (True)
Since both inequalities are satisfied, [tex]\((2, 5)\)[/tex] lies in the solution set.
### Point [tex]\((-2, -5)\)[/tex]
- For [tex]\(y > -3x + 3\)[/tex]:
[tex]\( -5 > -3(-2) + 3 \)[/tex]
[tex]\( -5 > 6 + 3 \)[/tex]
[tex]\( -5 > 9 \)[/tex] (False)
Since one of the inequalities is not satisfied, [tex]\((-2, -5)\)[/tex] does not lie in the solution set.
### Conclusion
The point that lies in the solution set of the given system of inequalities is [tex]\((2, 5)\)[/tex].
