Which of the following is a solution to this inequality?

[tex]\[ y \ \textless \ \frac{2}{3} x + 2 \][/tex]

A. [tex]\((0,3)\)[/tex]

B. [tex]\((-3,1)\)[/tex]

C. [tex]\((3,5)\)[/tex]

D. [tex]\((1,2)\)[/tex]



Answer :

To determine which points are solutions to the inequality [tex]\( y < \frac{2}{3}x + 2 \)[/tex], we need to check each point by substituting the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates into the inequality. Here is the step-by-step evaluation for each point:

### Point [tex]\( (0, 3) \)[/tex]
1. Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 3 \)[/tex] into the inequality:
[tex]\[ y < \frac{2}{3} \cdot 0 + 2 \][/tex]
2. Simplify the right side:
[tex]\[ 3 < 2 \][/tex]
This statement is false.

### Point [tex]\( (-3, 1) \)[/tex]
1. Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = 1 \)[/tex] into the inequality:
[tex]\[ y < \frac{2}{3} \cdot (-3) + 2 \][/tex]
2. Simplify the right side:
[tex]\[ 1 < \frac{2}{3} \cdot (-3) + 2 \][/tex]
[tex]\[ 1 < -2 + 2 \][/tex]
[tex]\[ 1 < 0 \][/tex]
This statement is false.

### Point [tex]\( (3, 5) \)[/tex]
1. Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 5 \)[/tex] into the inequality:
[tex]\[ y < \frac{2}{3} \cdot 3 + 2 \][/tex]
2. Simplify the right side:
[tex]\[ 5 < \frac{2}{3} \cdot 3 + 2 \][/tex]
[tex]\[ 5 < 2 + 2 \][/tex]
[tex]\[ 5 < 4 \][/tex]
This statement is false.

### Point [tex]\( (1, 2) \)[/tex]
1. Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = 2 \)[/tex] into the inequality:
[tex]\[ y < \frac{2}{3} \cdot 1 + 2 \][/tex]
2. Simplify the right side:
[tex]\[ 2 < \frac{2}{3} \cdot 1 + 2 \][/tex]
[tex]\[ 2 < \frac{2}{3} + 2 \][/tex]
[tex]\[ 2 < \frac{2}{3} + \frac{6}{3} \][/tex]
[tex]\[ 2 < \frac{8}{3} \][/tex]
[tex]\[ 2 < 2.67 \][/tex]
This statement is true.

### Conclusion
The point that is a solution to the inequality [tex]\( y < \frac{2}{3}x + 2 \)[/tex] is [tex]\( (1, 2) \)[/tex].