Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
$y > -2x - 7$
$y \geq x + 8$

Explanation:

Step1: Graph \( y > -2x - 7 \)

The boundary line is \( y = -2x - 7 \), which has a slope of \(-2\) and a y - intercept of \(-7\). Since the inequality is \( y > -2x - 7 \), we draw a dashed line (because the inequality is strict, \(>\)) and shade the region above the line. To find the direction of shading, we can test a point not on the line, for example, \((0,0)\). Substituting into the inequality: \( 0 > -2(0)-7\), \(0 > - 7\), which is true, so we shade the region that includes \((0,0)\) relative to the line \( y=-2x - 7\).

Step2: Graph \( y \geq x + 8 \)

The boundary line is \( y=x + 8\), with a slope of \(1\) and a y - intercept of \(8\). Since the inequality is \( y\geq x + 8\), we draw a solid line (because the inequality is non - strict, \(\geq\)) and shade the region above the line. Testing the point \((0,0)\): \(0\geq0 + 8\) is false, so we shade the region that does not include \((0,0)\) relative to the line \( y = x+8\).

Step3: Find the Intersection Region

The solution set of the system of inequalities is the region that is shaded by both inequalities. To find a point in the solution set, we can find the intersection of the two lines \( y=-2x - 7\) and \( y=x + 8\) by setting \(-2x-7=x + 8\). Solving for \(x\): \(-2x-x=8 + 7\), \(-3x=15\), \(x=- 5\). Substituting \(x = - 5\) into \(y=x + 8\), we get \(y=-5 + 8=3\). Now, we can choose a point in the overlapping region. Let's try \(x=-4\). For \(y > -2x-7\), when \(x = - 4\), \(y>-2(-4)-7=8 - 7 = 1\). For \(y\geq x + 8\), when \(x=-4\), \(y\geq-4 + 8 = 4\). So a point like \((-4,5)\) can be in the solution set. Let's check: For \(y > -2x-7\), \(5>-2(-4)-7=8 - 7 = 1\) (true). For \(y\geq x + 8\), \(5\geq-4 + 8 = 4\) (true).

Answer:

A point in the solution set is \((-4,5)\) (other valid points can also be chosen as long as they satisfy both inequalities). The graphing process involves drawing the dashed line for \(y > -2x - 7\) and shading above it, drawing the solid line for \(y\geq x + 8\) and shading above it, and the solution region is the intersection of the two shaded regions.