To determine which of the given points lies within the solution set of the inequality [tex]\( y < x^2 - 4x + 3 \)[/tex], we need to evaluate each point to see if it satisfies the inequality.
The inequality is:
[tex]\[ y < x^2 - 4x + 3 \][/tex]
Let's evaluate each point:
### Point [tex]\((-3, 0)\)[/tex]
First, we substitute [tex]\( x = -3 \)[/tex] into the expression [tex]\( x^2 - 4x + 3 \)[/tex]:
[tex]\[
(-3)^2 - 4(-3) + 3 = 9 + 12 + 3 = 24
\][/tex]
Next, we check if the [tex]\( y \)[/tex]-value satisfies the inequality:
[tex]\[
0 < 24
\][/tex]
This inequality is true, so the point [tex]\((-3, 0)\)[/tex] satisfies [tex]\( y < x^2 - 4x + 3 \)[/tex].
### Point [tex]\((2, -1)\)[/tex]
Next, we substitute [tex]\( x = 2 \)[/tex] into the expression [tex]\( x^2 - 4x + 3 \)[/tex]:
[tex]\[
2^2 - 4(2) + 3 = 4 - 8 + 3 = -1
\][/tex]
Next, we check if the [tex]\( y \)[/tex]-value satisfies the inequality:
[tex]\[
-1 < -1
\][/tex]
This inequality is not true since [tex]\(-1\)[/tex] is not less than [tex]\(-1\)[/tex], so the point [tex]\((2, -1)\)[/tex] does not satisfy the inequality [tex]\( y < x^2 - 4x + 3 \)[/tex].
### Point [tex]\((1, 0)\)[/tex]
Finally, we substitute [tex]\( x = 1 \)[/tex] into the expression [tex]\( x^2 - 4x + 3 \)[/tex]:
[tex]\[
1^2 - 4(1) + 3 = 1 - 4 + 3 = 0
\][/tex]
Next, we check if the [tex]\( y \)[/tex]-value satisfies the inequality:
[tex]\[
0 < 0
\][/tex]
This inequality is not true since [tex]\(0\)[/tex] is not less than [tex]\(0\)[/tex], so the point [tex]\((1, 0)\)[/tex] does not satisfy the inequality [tex]\( y < x^2 - 4x + 3 \)[/tex].
### Conclusion
Among the given points, the only point that satisfies the inequality [tex]\( y < x^2 - 4x + 3 \)[/tex] is [tex]\((-3, 0)\)[/tex].
