To determine which ordered pairs make both inequalities true, we need to test each pair against the given inequalities:
1. [tex]\(2x + 3y \leq 10\)[/tex]
2. [tex]\(x - y \geq -2\)[/tex]
Let's check each ordered pair step-by-step:
1. Pair: [tex]\((-5, 5)\)[/tex]
[tex]\[
\begin{align*}
2x + 3y & = 2(-5) + 3(5) = -10 + 15 = 5 & \text{(satisfies } 2x + 3y \leq 10\text{)} \\
x - y & = -5 - 5 = -10 & \text{(does not satisfy } x - y \geq -2\text{)}
\end{align*}
\][/tex]
This pair does not make both inequalities true.
2. Pair: [tex]\((0, 3)\)[/tex]
[tex]\[
\begin{align*}
2x + 3y & = 2(0) + 3(3) = 0 + 9 = 9 & \text{(satisfies } 2x + 3y \leq 10\text{)} \\
x - y & = 0 - 3 = -3 & \text{(does not satisfy } x - y \geq -2\text{)}
\end{align*}
\][/tex]
This pair does not make both inequalities true.
3. Pair: [tex]\((0, -2)\)[/tex]
[tex]\[
\begin{align*}
2x + 3y & = 2(0) + 3(-2) = 0 - 6 = -6 & \text{(satisfies } 2x + 3y \leq 10\text{)} \\
x - y & = 0 - (-2) = 0 + 2 = 2 & \text{(satisfies } x - y \geq -2\text{)}
\end{align*}
\][/tex]
This pair makes both inequalities true.
4. Pair: [tex]\((1, 1)\)[/tex]
[tex]\[
\begin{align*}
2x + 3y & = 2(1) + 3(1) = 2 + 3 = 5 & \text{(satisfies } 2x + 3y \leq 10\text{)} \\
x - y & = 1 - 1 = 0 & \text{(satisfies } x - y \geq -2\text{)}
\end{align*}
\][/tex]
This pair makes both inequalities true.
5. Pair: [tex]\((3, -4)\)[/tex]
[tex]\[
\begin{align*}
2x + 3y & = 2(3) + 3(-4) = 6 - 12 = -6 & \text{(satisfies } 2x + 3y \leq 10\text{)} \\
x - y & = 3 - (-4) = 3 + 4 = 7 & \text{(satisfies } x - y \geq -2\text{)}
\end{align*}
\][/tex]
This pair makes both inequalities true.
Therefore, the ordered pairs that make both inequalities true are:
[tex]\(\boxed{(0, -2), (1, 1), (3, -4)}\)[/tex]
