Question

explaining the relationship between a point and a sys
determine the relationship between the point (1, -5) and
the given system of inequalities.
y ≤ 3x + 2
y > -2x - 3
explain your answer both algebraically and graphically.

Explanation:

Step1: Check first inequality

Substitute \(x = 1\), \(y=-5\) into \(y\leq3x + 2\).
Left side: \(y=-5\), Right side: \(3(1)+2 = 5\).
Since \(-5\leq5\), the point satisfies \(y\leq3x + 2\).

Step2: Check second inequality

Substitute \(x = 1\), \(y = -5\) into \(y>-2x-3\).
Right side: \(-2(1)-3=-5\).
We need to check if \(-5>-5\), but \(-5=-5\), so the point does not satisfy \(y>-2x - 3\).

Step3: Graphical explanation

1. For \(y\leq3x + 2\), the line \(y = 3x+2\) has a slope of 3 and y - intercept 2. The region \(y\leq3x + 2\) is below (including the line) this line. The point \((1,-5)\) is below this line.
2. For \(y>-2x-3\), the line \(y=-2x - 3\) has a slope of - 2 and y - intercept - 3. The region \(y>-2x-3\) is above this line. The point \((1,-5)\) lies on the line \(y=-2x - 3\) (since when \(x = 1\), \(y=-2(1)-3=-5\)), so it is not in the region \(y>-2x-3\).

Since the point must satisfy both inequalities to be in the solution set of the system, and it fails to satisfy the second inequality, the point \((1,-5)\) is not in the solution set of the system of inequalities.

Answer:

The point \((1, - 5)\) satisfies \(y\leq3x + 2\) (algebraically and lies below the line \(y = 3x+2\) graphically) but does not satisfy \(y>-2x - 3\) (algebraically, it lies on the line \(y=-2x - 3\) graphically and not above it). So, the point \((1,-5)\) is not in the solution set of the given system of inequalities.