To determine which of the given points lie in the solution set of the system of inequalities:
[tex]\[
\begin{array}{l}
y \leq x - 5 \\
y \geq -x - 4
\end{array}
\][/tex]
we will check each point one by one against both inequalities.
1. For the point [tex]\((-5, 2)\)[/tex]:
- First inequality: [tex]\(2 \leq -5 - 5\)[/tex]
Simplifying, we get [tex]\(2 \leq -10\)[/tex], which is false.
Since [tex]\((-5, 2)\)[/tex] does not satisfy the first inequality, there is no need to check the second inequality. Therefore, [tex]\((-5, 2)\)[/tex] does not lie in the solution set.
2. For the point [tex]\((5, -2)\)[/tex]:
- First inequality: [tex]\(-2 \leq 5 - 5\)[/tex]
Simplifying, we get [tex]\(-2 \leq 0\)[/tex], which is true.
- Second inequality: [tex]\(-2 \geq -5 - 4\)[/tex]
Simplifying, we get [tex]\(-2 \geq -9\)[/tex], which is true.
Since [tex]\((5, -2)\)[/tex] satisfies both inequalities, this point does lie in the solution set.
3. For the point [tex]\((-5, -2)\)[/tex]:
- First inequality: [tex]\(-2 \leq -5 - 5\)[/tex]
Simplifying, we get [tex]\(-2 \leq -10\)[/tex], which is false.
Since [tex]\((-5, -2)\)[/tex] does not satisfy the first inequality, there is no need to check the second inequality. Therefore, [tex]\((-5, -2)\)[/tex] does not lie in the solution set.
4. For the point [tex]\((5, 2)\)[/tex]:
- First inequality: [tex]\(2 \leq 5 - 5\)[/tex]
Simplifying, we get [tex]\(2 \leq 0\)[/tex], which is false.
Since [tex]\((5, 2)\)[/tex] does not satisfy the first inequality, there is no need to check the second inequality. Therefore, [tex]\((5, 2)\)[/tex] does not lie in the solution set.
Summarizing the points:
- [tex]\((-5, 2)\)[/tex]: does not lie in the solution set.
- [tex]\((5, -2)\)[/tex]: lies in the solution set.
- [tex]\((-5, -2)\)[/tex]: does not lie in the solution set.
- [tex]\((5, 2)\)[/tex]: does not lie in the solution set.
Thus, the only point that lies in the solution set of the given system of inequalities is [tex]\((5, -2)\)[/tex].
