determine whether or not each of the following point in a solution of the enequality y<2xsquared+3x-5  2.( 1,8)=?   3.(-5,10) =?     4.(3,6) =?      5.(-3,4) = ?       6.(2,9)= ?                                                                                                                                                                             example: 1.a(-1,6)  solution:  y<2xsquared+3x-5     6<2(-1)xsquared+3(-1)-5  6<2(-1)-3-5    6<2-8          6<-6        false                                                    

To determine whether a point is a solution of the inequality \( y < 2x^2 + 3x - 5 \), we will plug in the coordinates of each point \((x, y)\) into the inequality and check whether the inequality holds.

### Given Inequality:
\[ y < 2x^2 + 3x - 5 \]

### Example:
For point \((-1, 6)\):
\[
y < 2x^2 + 3x - 5
\]
Substituting \((-1, 6)\) into the inequality:
\[
6 < 2(-1)^2 + 3(-1) - 5
\]
\[
6 < 2(1) - 3 - 5
\]
\[
6 < 2 - 3 - 5
\]
\[
6 < -6 \quad \text{(False)}
\]
So, point \((-1, 6)\) is **not** a solution.

### Now let's evaluate the other points:

### 2. Point \( (1, 8) \):
Substitute \( x = 1 \) and \( y = 8 \) into the inequality:
\[
8 < 2(1)^2 + 3(1) - 5
\]
\[
8 < 2(1) + 3 - 5
\]
\[
8 < 2 + 3 - 5
\]
\[
8 < 0 \quad \text{(False)}
\]
So, point \( (1, 8) \) is **not** a solution.

### 3. Point \( (-5, 10) \):
Substitute \( x = -5 \) and \( y = 10 \) into the inequality:
\[
10 < 2(-5)^2 + 3(-5) - 5
\]
\[
10 < 2(25) - 15 - 5
\]
\[
10 < 50 - 15 - 5
\]
\[
10 < 30 \quad \text{(True)}
\]
So, point \( (-5, 10) \) **is** a solution.

### 4. Point \( (3, 6) \):
Substitute \( x = 3 \) and \( y = 6 \) into the inequality:
\[
6 < 2(3)^2 + 3(3) - 5
\]
\[
6 < 2(9) + 9 - 5
\]
\[
6 < 18 + 9 - 5
\]
\[
6 < 22 \quad \text{(True)}
\]
So, point \( (3, 6) \) **is** a solution.

### 5. Point \( (-3, 4) \):
Substitute \( x = -3 \) and \( y = 4 \) into the inequality:
\[
4 < 2(-3)^2 + 3(-3) - 5
\]
\[
4 < 2(9) - 9 - 5
\]
\[
4 < 18 - 9 - 5
\]
\[
4 < 4 \quad \text{(False)}
\]
So, point \( (-3, 4) \) is **not** a solution.

### 6. Point \( (2, 9) \):
Substitute \( x = 2 \) and \( y = 9 \) into the inequality:
\[
9 < 2(2)^2 + 3(2) - 5
\]
\[
9 < 2(4) + 6 - 5
\]
\[
9 < 8 + 6 - 5
\]
\[
9 < 9 \quad \text{(False)}
\]
So, point \( (2, 9) \) is **not** a solution.

### Summary of Results:
1. \( (1, 8) \): False
2. \( (-5, 10) \): True
3. \( (3, 6) \): True
4. \( (-3, 4) \): False
5. \( (2, 9) \): False

Do you want more details or have any questions?

Here are 5 related questions:
1. How do we solve a quadratic inequality like \( y < 2x^2 + 3x - 5 \)?
2. What is the significance of the boundary curve \( y = 2x^2 + 3x - 5 \)?
3. How can we graphically check the solutions to this inequality?
4. What does it mean for a point to lie "inside" or "outside" the solution region?
5. How would the solution change if the inequality were \( y \leq 2x^2 + 3x - 5 \)?

**Tip:** Always check if your quadratic inequality has been simplified properly before substituting points.

Determine whether or not each of the following point is a solution of the inequality y < 2x^2 + 3x - 5: (1,8), (-5,10), (3,6), (-3,4), (2,9).

Explore how to determine whether given points, such as (1,8) or (-5,10), satisfy the quadratic inequality y < 2x^2 + 3x - 5. Learn key algebraic concepts like substitution, quadratic inequalities, and step-by-step solutions. This problem is suitable for students in Grades 8-10.

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