Question

solving systems of linear inequalities
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: Test first inequality

Substitute $x=1, y=-5$ into $y \leq 3x + 2$:

$$\begin{align*} -5 &\leq 3(1) + 2 \\ -5 &\leq 5 \end{align*}$$

This is true.

Step2: Test second inequality

Substitute $x=1, y=-5$ into $y > -2x - 3$:

$$\begin{align*} -5 &> -2(1) - 3 \\ -5 &> -5 \end{align*}$$

This is false.

Step3: Analyze line of second inequality

The second inequality is $y > -2x - 3$, which uses a dashed line (since it's strict, no equality). Substitute $x=1$ into the line equation $y=-2x-3$:
$y=-2(1)-3=-5$, so the point lies on this dashed line.

Step4: Evaluate solution set

Since the point fails the second inequality, it is not in the system's solution set, and does not lie in the shaded area of both inequalities.

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.