To determine which ordered pairs make both inequalities true, let's consider the following two inequalities:
1. [tex]\(2x + 3y < 10\)[/tex]
2. [tex]\(x - y > -4\)[/tex]
We will evaluate each ordered pair [tex]\((x, y)\)[/tex] to see if it satisfies both inequalities.
1. For [tex]\((-5, 5)\)[/tex]:
- First inequality: [tex]\(2(-5) + 3(5) = -10 + 15 = 5\)[/tex], which is less than 10.
- Second inequality: [tex]\(-5 - 5 = -10\)[/tex], which is not greater than -4.
- Therefore, [tex]\((-5, 5)\)[/tex] does not satisfy both inequalities.
2. For [tex]\((0, 3)\)[/tex]:
- First inequality: [tex]\(2(0) + 3(3) = 0 + 9 = 9\)[/tex], which is less than 10.
- Second inequality: [tex]\(0 - 3 = -3\)[/tex], which is greater than -4.
- Therefore, [tex]\((0, 3)\)[/tex] satisfies both inequalities.
3. For [tex]\((0, -2)\)[/tex]:
- First inequality: [tex]\(2(0) + 3(-2) = 0 - 6 = -6\)[/tex], which is less than 10.
- Second inequality: [tex]\(0 - (-2) = 0 + 2 = 2\)[/tex], which is greater than -4.
- Therefore, [tex]\((0, -2)\)[/tex] satisfies both inequalities.
4. For [tex]\((1, 1)\)[/tex]:
- First inequality: [tex]\(2(1) + 3(1) = 2 + 3 = 5\)[/tex], which is less than 10.
- Second inequality: [tex]\(1 - 1 = 0\)[/tex], which is greater than -4.
- Therefore, [tex]\((1, 1)\)[/tex] satisfies both inequalities.
5. For [tex]\((3, -4)\)[/tex]:
- First inequality: [tex]\(2(3) + 3(-4) = 6 - 12 = -6\)[/tex], which is less than 10.
- Second inequality: [tex]\(3 - (-4) = 3 + 4 = 7\)[/tex], which is greater than -4.
- Therefore, [tex]\((3, -4)\)[/tex] satisfies both inequalities.
In conclusion, the ordered pairs that make both inequalities true are:
[tex]\[
(0, 3), (0, -2), (1, 1), (3, -4)
\][/tex]
