To solve the inequality [tex]\( y > 5x - 10 \)[/tex], we need to understand what points in the coordinate plane satisfy this inequality. Let's examine specific points to see if they satisfy the inequality.
We take some example points [tex]\((x, y)\)[/tex]:
1. Point (0, 0):
- Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality.
- We get [tex]\( 0 > 5 \cdot 0 - 10 \)[/tex].
- This simplifies to [tex]\( 0 > -10 \)[/tex].
- This statement is true.
2. Point (2, 0):
- Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality.
- We get [tex]\( 0 > 5 \cdot 2 - 10 \)[/tex].
- This simplifies to [tex]\( 0 > 10 - 10 \)[/tex], which further simplifies to [tex]\( 0 > 0 \)[/tex].
- This statement is false.
3. Point (-1, 0):
- Substitute [tex]\(x = -1\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality.
- We get [tex]\( 0 > 5 \cdot (-1) - 10 \)[/tex].
- This simplifies to [tex]\( 0 > -5 - 10 \)[/tex], which further simplifies to [tex]\( 0 > -15 \)[/tex].
- This statement is true.
4. Point (3, 10):
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = 10\)[/tex] into the inequality.
- We get [tex]\( 10 > 5 \cdot 3 - 10 \)[/tex].
- This simplifies to [tex]\( 10 > 15 - 10 \)[/tex], which further simplifies to [tex]\( 10 > 5 \)[/tex].
- This statement is true.
So, the results for these points are as follows:
- (0, 0) satisfies the inequality: True
- (2, 0) satisfies the inequality: False
- (-1, 0) satisfies the inequality: True
- (3, 10) satisfies the inequality: True
In summary, the points [tex]\((0, 0)\)[/tex], [tex]\((-1, 0)\)[/tex], and [tex]\((3, 10)\)[/tex] satisfy the inequality [tex]\( y > 5x - 10 \)[/tex], while the point [tex]\((2, 0)\)[/tex] does not.
