To determine which of the inequalities the point [tex]\((0,0)\)[/tex] satisfies, we need to substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into each inequality and check if the resulting statement is true.
Let's analyze each inequality one by one:
### Inequality A: [tex]\( y + 7 < 2x - 6 \)[/tex]
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex]:
[tex]\[ 0 + 7 < 2(0) - 6 \][/tex]
[tex]\[ 7 < -6 \][/tex]
This statement is false.
### Inequality B: [tex]\( y - 6 < 2x - 7 \)[/tex]
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex]:
[tex]\[ 0 - 6 < 2(0) - 7 \][/tex]
[tex]\[ -6 < -7 \][/tex]
This statement is false.
### Inequality C: [tex]\( y - 7 < 2x - 6 \)[/tex]
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex]:
[tex]\[ 0 - 7 < 2(0) - 6 \][/tex]
[tex]\[ -7 < -6 \][/tex]
This statement is true.
### Inequality D: [tex]\( y + 7 < 2x + 6 \)[/tex]
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex]:
[tex]\[ 0 + 7 < 2(0) + 6 \][/tex]
[tex]\[ 7 < 6 \][/tex]
This statement is false.
So, after evaluating the inequalities with the point [tex]\((0,0)\)[/tex], we find that the point [tex]\((0,0)\)[/tex] is a solution only to inequality [tex]\( \text{C}. \)[/tex]
