To determine which inequalities can pair with [tex]\( x + y \geq 22 \)[/tex] to create a system representing the given solution, we should evaluate each inequality.
1. Evaluate [tex]\( 8x + 12y > 240 \)[/tex]:
To check if this inequality can pair with [tex]\( x + y \geq 22 \)[/tex], we consider a test point that satisfies [tex]\( x + y \geq 22 \)[/tex]. Using the point [tex]\( (30, 10) \)[/tex]:
[tex]\[
8(30) + 12(10) = 240 + 120 = 360
\][/tex]
Since [tex]\( 360 > 240 \)[/tex], [tex]\( (30, 10) \)[/tex] satisfies [tex]\( 8x + 12y > 240 \)[/tex]. Hence, the inequality [tex]\( 8x + 12y > 240 \)[/tex] works with [tex]\( x + y \geq 22 \)[/tex].
2. Evaluate [tex]\( 12x + 8y > 240 \)[/tex]:
Let's use the same test point [tex]\( (30, 10) \)[/tex]:
[tex]\[
12(30) + 8(10) = 360 + 80 = 440
\][/tex]
Since [tex]\( 440 > 240 \)[/tex], [tex]\( (30, 10) \)[/tex] satisfies [tex]\( 12x + 8y > 240 \)[/tex]. Thus, the inequality [tex]\( 12x + 8y > 240 \)[/tex] also works with [tex]\( x + y \geq 22 \)[/tex].
3. Evaluate [tex]\( 8x + 12y < 240 \)[/tex]:
Consider the test point [tex]\( (10, 10) \)[/tex]:
[tex]\[
8(10) + 12(10) = 80 + 120 = 200
\][/tex]
Since [tex]\( 200 < 240 \)[/tex], [tex]\( (10, 10) \)[/tex] satisfies [tex]\( 8x + 12y < 240 \)[/tex]. Thus, the inequality [tex]\( 8x + 12y < 240 \)[/tex] works with [tex]\( x + y \geq 22 \)[/tex].
4. Evaluate [tex]\( 12x + 8y < 240 \)[/tex]:
Using the same test point [tex]\( (10, 10) \)[/tex]:
[tex]\[
12(10) + 8(10) = 120 + 80 = 200
\][/tex]
Since [tex]\( 200 < 240 \)[/tex], [tex]\( (10, 10) \)[/tex] satisfies [tex]\( 12x + 8y < 240 \)[/tex]. Thus, this inequality [tex]\( 12x + 8y < 240 \)[/tex] works with [tex]\( x + y \geq 22 \)[/tex].
Thus, all the provided inequalities, [tex]\( 8x + 12y > 240 \)[/tex], [tex]\( 12x + 8y > 240 \)[/tex], [tex]\( 8x + 12y < 240 \)[/tex], and [tex]\( 12x + 8y < 240 \)[/tex], can be appropriately paired with [tex]\( x + y \geq 22 \)[/tex] to form a compatible system of inequalities reflecting the given solutions.
