Let's solve this problem step-by-step to determine which point is not included in the solution set for the inequality.
Assume we have the inequality [tex]\( y < 2x + 3 \)[/tex].
We will test each point against this inequality:
1. Point [tex]\( (0, 6) \)[/tex]:
- Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 6 \)[/tex] into the inequality [tex]\( y < 2x + 3 \)[/tex]:
[tex]\[
6 < 2(0) + 3 \implies 6 < 3
\][/tex]
- This statement is false because 6 is not less than 3. Therefore, the point [tex]\( (0, 6) \)[/tex] does not satisfy the inequality.
2. Point [tex]\( (1, 5) \)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = 5 \)[/tex] into the inequality [tex]\( y < 2x + 3 \)[/tex]:
[tex]\[
5 < 2(1) + 3 \implies 5 < 5
\][/tex]
- This statement is false because 5 is not less than 5. However, [tex]\( 5 = 5 \)[/tex], and we need [tex]\( y \)[/tex] to be strictly less than [tex]\( 2x + 3 \)[/tex]. So, the point [tex]\( (1, 5) \)[/tex] does not satisfy the inequality.
3. Point [tex]\( (2, 4) \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 4 \)[/tex] into the inequality [tex]\( y < 2x + 3 \)[/tex]:
[tex]\[
4 < 2(2) + 3 \implies 4 < 7
\][/tex]
- This statement is true because 4 is less than 7. Therefore, the point [tex]\( (2, 4) \)[/tex] satisfies the inequality.
4. Point [tex]\( (3, 2) \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 2 \)[/tex] into the inequality [tex]\( y < 2x + 3 \)[/tex]:
[tex]\[
2 < 2(3) + 3 \implies 2 < 9
\][/tex]
- This statement is true because 2 is less than 9. Therefore, the point [tex]\( (3, 2) \)[/tex] satisfies the inequality.
Based on the above tests, the point that does not satisfy the inequality [tex]\( y < 2x + 3 \)[/tex] is [tex]\( (0, 6) \)[/tex]. Therefore, the point that is not included in the solution set for the inequality is [tex]\( (0, 6) \)[/tex].
