To solve the inequality [tex]\( y > \frac{1}{3} x - 2 \)[/tex], we need to determine which points in the coordinate plane satisfy this inequality.
1. Understand the Equation of the Line:
The inequality is based on the line [tex]\( y = \frac{1}{3} x - 2 \)[/tex]. This line divides the plane into two regions:
- Points above the line satisfy [tex]\( y > \frac{1}{3} x - 2 \)[/tex].
- Points below the line do not satisfy the inequality.
2. Testing Points to the Inequality:
Let's test a few points to see if they satisfy the inequality:
- Point (0, 0):
[tex]\[
y = 0 \quad \text{and} \quad x = 0
\][/tex]
Substitute into the inequality:
[tex]\[
0 > \frac{1}{3}(0) - 2 \implies 0 > -2
\][/tex]
This is true. So, the point (0, 0) satisfies the inequality.
- Point (3, 0):
[tex]\[
y = 0 \quad \text{and} \quad x = 3
\][/tex]
Substitute into the inequality:
[tex]\[
0 > \frac{1}{3}(3) - 2 \implies 0 > 1 - 2 \implies 0 > -1
\][/tex]
This is true. So, the point (3, 0) satisfies the inequality.
- Point (6, 2):
[tex]\[
y = 2 \quad \text{and} \quad x = 6
\][/tex]
Substitute into the inequality:
[tex]\[
2 > \frac{1}{3}(6) - 2 \implies 2 > 2 - 2 \implies 2 > 0
\][/tex]
This is true. So, the point (6, 2) satisfies the inequality.
3. Conclusion:
The points (0, 0), (3, 0), and (6, 2) all satisfy the inequality [tex]\( y > \frac{1}{3} x - 2 \)[/tex]. Therefore, any point above the line defined by [tex]\( y = \frac{1}{3} x - 2 \)[/tex] will satisfy the given inequality.
By systematically testing points and confirming they satisfy the inequality, we can conclude that the solution consists of all points above the line [tex]\( y = \frac{1}{3} x - 2 \)[/tex]. This approach can be used to verify other points as well.
