determine the relationship between the point (1, -5) and the given system of inequalities.\ny ≤ 3x + 2\ny >…

determine the relationship between the point (1, -5) and the given system of inequalities.\ny ≤ 3x + 2\ny > -2x - 3\nexplain your answer both algebraically and graphically.

determine the relationship between the point (1, -5) and the given system of inequalities.\ny ≤ 3x + 2\ny > -2x - 3\nexplain your answer both algebraically and graphically.

Answer

Answer:

The point (1, - 5) satisfies the system of inequalities.

Explanation:

Step1: Substitute into first - inequality

Substitute (x = 1) and (y=-5) into (y\leq3x + 2). (-5\leq3\times1+2), (-5\leq3 + 2), (-5\leq5) (True).

Step2: Substitute into second - inequality

Substitute (x = 1) and (y = - 5) into (y>-2x-3). (-5>-2\times1-3), (-5>-2 - 3), (-5>-5) (False). But if we correct the second - step calculation: (-5>-2\times1 - 3) gives (-5>-2-3=-5) which is wrong. The correct second - step: Substitute (x = 1) and (y=-5) into (y>-2x - 3), we get (-5>-2\times1-3=-2 - 3=-5) is incorrect. Let's start over.

Step1: Substitute into first inequality

Substitute (x = 1) and (y=-5) into (y\leq3x + 2). We have (-5\leq3\times1+2=5), which is True.

Step2: Substitute into second inequality

Substitute (x = 1) and (y=-5) into (y>-2x-3). (y=-5), and (-2x - 3=-2\times1-3=-5). Since (-5=-5) and the inequality is (y >-2x - 3), this is False. Graphically: The inequality (y\leq3x + 2) represents the region below (and including) the line (y = 3x+2). The line (y = 3x+2) has a y - intercept of 2 and a slope of 3. The inequality (y>-2x-3) represents the region above the line (y=-2x - 3) which has a y - intercept of - 3 and a slope of - 2. When we plot the point ((1,-5)), we can see that it lies on the boundary of (y=-2x - 3) (since (-5=-2\times1-3)) and below (y = 3x+2) (since (-5<3\times1 + 2)). Since the second inequality is (y>-2x - 3) (strict inequality), the point ((1,-5)) does not satisfy the system of inequalities. The point (1, - 5) does not satisfy the system of inequalities.

Step1: Substitute into first inequality

Substitute (x = 1) and (y=-5) into (y\leq3x + 2). (y=-5), (3x + 2=3\times1+2 = 5), so (-5\leq5) (True).

Step2: Substitute into second inequality

Substitute (x = 1) and (y=-5) into (y>-2x-3). (-2x-3=-2\times1-3=-5), and the inequality is (y >-2x - 3). Since (-5=-5) and not (-5>-5), this is False. So the point (1, - 5) does not satisfy the system of inequalities.