Certainly! Let's solve the system of inequalities step by step and determine which ordered pair is a solution.
Given inequalities:
1. [tex]\( y \leq -x^2 + 8x \)[/tex]
2. [tex]\( y > x^2 - 3 \)[/tex]
We need to check each of the given points to see if they satisfy both inequalities.
Option A: [tex]\((-1, 4)\)[/tex]
- For the first inequality [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\( x = -1 \)[/tex]
[tex]\( -(-1)^2 + 8(-1) = -1 - 8 = -9 \)[/tex]
So, [tex]\( y \leq -9 \)[/tex]
Given [tex]\( y = 4 \)[/tex], which does not satisfy [tex]\( y \leq -9 \)[/tex]
Therefore, [tex]\((-1, 4)\)[/tex] does not satisfy the first inequality.
Option B: [tex]\( (0, 9) \)[/tex]
- For the first inequality [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\( x = 0 \)[/tex]
[tex]\( -0^2 + 8(0) = 0 \)[/tex]
So, [tex]\( y \leq 0 \)[/tex]
Given [tex]\( y = 9 \)[/tex], which does not satisfy [tex]\( y \leq 0 \)[/tex]
Therefore, [tex]\( (0, 9) \)[/tex] does not satisfy the first inequality.
Option C: [tex]\( (2, 7) \)[/tex]
- For the first inequality [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\( x = 2 \)[/tex]
[tex]\( -(2)^2 + 8(2) = -4 + 16 = 12 \)[/tex]
So, [tex]\( y \leq 12 \)[/tex]
Given [tex]\( y = 7 \)[/tex], which satisfies [tex]\( y \leq 12 \)[/tex]
- For the second inequality [tex]\( y > x^2 - 3 \)[/tex]:
[tex]\( x = 2 \)[/tex]
[tex]\( (2)^2 - 3 = 4 - 3 = 1 \)[/tex]
So, [tex]\( y > 1 \)[/tex]
Given [tex]\( y = 7 \)[/tex], which satisfies [tex]\( y > 1 \)[/tex]
Therefore, [tex]\( (2, 7) \)[/tex] satisfies both inequalities.
Option D: [tex]\( (4, 3) \)[/tex]
- For the first inequality [tex]\( y \leq -x^2 + 8x \)[/tex]:
[tex]\( x = 4 \)[/tex]
[tex]\( -(4)^2 + 8(4) = -16 + 32 = 16 \)[/tex]
So, [tex]\( y \leq 16 \)[/tex]
Given [tex]\( y = 3 \)[/tex], which satisfies [tex]\( y \leq 16 \)[/tex]
- For the second inequality [tex]\( y > x^2 - 3 \)[/tex]:
[tex]\( x = 4 \)[/tex]
[tex]\( (4)^2 - 3 = 16 - 3 = 13 \)[/tex]
So, [tex]\( y > 13 \)[/tex]
Given [tex]\( y = 3 \)[/tex], which does not satisfy [tex]\( y > 13 \)[/tex]
Therefore, [tex]\( (4, 3) \)[/tex] does not satisfy the second inequality.
After checking all options, only [tex]\( (2, 7) \)[/tex] satisfies both inequalities.
Thus, the ordered pair that is a solution to the given system of inequalities is [tex]\(\boxed{(2, 7)}\)[/tex].
