To determine which of the given inequalities the point \((0,0)\) is a solution, we need to substitute \(x = 0\) and \(y = 0\) into each inequality and check if the resulting statements are true. Let's evaluate each inequality one by one:
Inequality A: \(y + 4 < 3x + 1\)
Substitute \(x = 0\) and \(y = 0\):
[tex]\[ 0 + 4 < 3(0) + 1 \][/tex]
[tex]\[ 4 < 1 \][/tex]
This statement is False.
Inequality B: \(y - 1 < 3x - 4\)
Substitute \(x = 0\) and \(y = 0\):
[tex]\[ 0 - 1 < 3(0) - 4 \][/tex]
[tex]\[ -1 < -4 \][/tex]
This statement is False.
Inequality C: \(y - 4 < 3x - 1\)
Substitute \(x = 0\) and \(y = 0\):
[tex]\[ 0 - 4 < 3(0) - 1 \][/tex]
[tex]\[ -4 < -1 \][/tex]
This statement is True.
Inequality D: \(y + 4 < 3x - 1\)
Substitute \(x = 0\) and \(y = 0\):
[tex]\[ 0 + 4 < 3(0) - 1 \][/tex]
[tex]\[ 4 < -1 \][/tex]
This statement is False.
So, the point \((0,0)\) is a solution only to inequality:
C. [tex]\(y - 4 < 3x - 1\)[/tex]
