Question

determine the relationship between the point (1, -5) and the given system of inequalities. choose three correct answers.
$y \leq 3x + 2$
$y > -2x - 3$

• the point (1, -5) satisfies the first inequality but not the second inequality.
• the point (1, -5) lies on the dashed line of the second inequality, which is not inclusive.
• the point (1, -5) is not in the solution set of the system of inequalities.
• the point (1, -5) is in the solution set of the system of inequalities.
• the point (1, -5) lies in the shaded area of both inequalities.

Explanation:

Step1: Check first inequality

Substitute \(x = 1\), \(y=-5\) into \(y\leq3x + 2\):
\(-5\leq3(1)+2\)
\(-5\leq5\), which is true.

Step2: Check second inequality

Substitute \(x = 1\), \(y = -5\) into \(y>-2x - 3\):
\(-5>-2(1)-3\)
\(-5>-5\), which is false (since \(-5\) is not greater than \(-5\)).

Analyze each option:

1. "The point \((1, -5)\) satisfies the first inequality but not the second inequality." → True (from Step1 and Step2).
2. "The point \((1, -5)\) lies on the dashed line of the second inequality, which is not inclusive." → The second inequality is \(y > -2x - 3\) (dashed line). At \(x = 1\), \(y=-5\) is equal to \(-2(1)-3=-5\), so it lies on the dashed line. The inequality is strict (\(>\)), so the line is not inclusive. → True.
3. "The point \((1, -5)\) is not in the solution set of the system of inequalities." → A system’s solution must satisfy both inequalities. The point fails the second inequality, so it is not in the solution set. → True.
4. "The point \((1, -5)\) is in the solution set of the system of inequalities." → False (fails the second inequality).
5. "The point \((1, -5)\) lies in the shaded area of both inequalities." → False (fails the second inequality, so not in both shaded areas).

Answer:

• The point \((1, -5)\) satisfies the first inequality but not the second inequality.
• The point \((1, -5)\) lies on the dashed line of the second inequality, which is not inclusive.
• The point \((1, -5)\) is not in the solution set of the system of inequalities.