To determine which ordered pairs [tex]\((x, y)\)[/tex] satisfy the inequality [tex]\(y > x^2 + 3x - 4\)[/tex], we need to check each pair by plugging the [tex]\(x\)[/tex] value into the quadratic expression [tex]\(x^2 + 3x - 4\)[/tex] and then comparing [tex]\(y\)[/tex] to the resulting value.
Let's check each pair step-by-step:
1. Pair [tex]\((0, 0)\)[/tex]:
[tex]\[
y = 0, \quad x = 0
\][/tex]
Substitute [tex]\(x = 0\)[/tex] into [tex]\(x^2 + 3x - 4\)[/tex]:
[tex]\[
0^2 + 3 \cdot 0 - 4 = -4
\][/tex]
Now, compare [tex]\(y\)[/tex] with [tex]\(-4\)[/tex]:
[tex]\[
0 > -4 \quad \text{(True)}
\][/tex]
Therefore, [tex]\((0, 0)\)[/tex] satisfies the inequality.
2. Pair [tex]\((2, 1)\)[/tex]:
[tex]\[
y = 1, \quad x = 2
\][/tex]
Substitute [tex]\(x = 2\)[/tex] into [tex]\(x^2 + 3x - 4\)[/tex]:
[tex]\[
2^2 + 3 \cdot 2 - 4 = 4 + 6 - 4 = 6
\][/tex]
Now, compare [tex]\(y\)[/tex] with 6:
[tex]\[
1 > 6 \quad \text{(False)}
\][/tex]
Therefore, [tex]\((2, 1)\)[/tex] does not satisfy the inequality.
3. Pair [tex]\((-2, -1)\)[/tex]:
[tex]\[
y = -1, \quad x = -2
\][/tex]
Substitute [tex]\(-2\)[/tex] into [tex]\(x^2 + 3x - 4\)[/tex]:
[tex]\[
(-2)^2 + 3 \cdot (-2) - 4 = 4 - 6 - 4 = -6
\][/tex]
Now, compare [tex]\(y\)[/tex] with [tex]\(-6\)[/tex]:
[tex]\[
-1 > -6 \quad \text{(True)}
\][/tex]
Therefore, [tex]\((-2, -1)\)[/tex] satisfies the inequality.
4. Pair [tex]\((-5, -1)\)[/tex]:
[tex]\[
y = -1, \quad x = -5
\][/tex]
Substitute [tex]\(-5\)[/tex] into [tex]\(x^2 + 3x - 4\)[/tex]:
[tex]\[
(-5)^2 + 3 \cdot (-5) - 4 = 25 - 15 - 4 = 6
\][/tex]
Now, compare [tex]\(y\)[/tex] with 6:
[tex]\[
-1 > 6 \quad \text{(False)}
\][/tex]
Therefore, [tex]\((-5, -1)\)[/tex] does not satisfy the inequality.
After evaluating each pair, the pairs that satisfy the inequality [tex]\(y > x^2 + 3x - 4\)[/tex] are:
[tex]\[
(0, 0) \quad \text{and} \quad (-2, -1)
\][/tex]
Thus, the selected pairs are [tex]\((0, 0)\)[/tex] and [tex]\((-2, -1)\)[/tex].
