Question

1. which inequality when graphed with $x + y > 6$, has the solution shown in the graph? (graph with points (3,4) and (-5,-3), and options: a) $7x - 8y \leq -11$; b) $7x - 8y \geq -11$; c) $8x - 7y \leq -11$; d) $8x - 7y \leq -11$)

Explanation:

Step1: Identify the intersection point

The two lines intersect at \((3, 4)\). We can use this point to test the inequalities. Also, we can find the equation of the other line (not \(x + y>6\)) by using the two points \((-5, -3)\) and \((3, 4)\).

Step2: Find the slope of the other line

The slope \(m\) between \((-5, -3)\) and \((3, 4)\) is \(m=\frac{4 - (-3)}{3 - (-5)}=\frac{7}{8}\). Using point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(3,4)\), we get \(y - 4=\frac{7}{8}(x - 3)\). Multiply through by 8: \(8y-32 = 7x - 21\), which simplifies to \(7x-8y=- 11\).

Step3: Determine the inequality sign

The shaded region (solution) is on one side of the line \(7x - 8y=-11\). Let's test a point in the shaded region, say \((0,0)\) (wait, no, \((0,0)\) is not in the shaded region. Let's take \((3,4)\) or another point. Wait, the line \(x + y>6\) and the other line. Let's test the point \((0,7)\) (which is in the shaded region). For the line \(7x - 8y=-11\), plug \(x = 0,y = 7\): \(7(0)-8(7)=-56\). We want to see if \(-56\leq - 11\) (since \(-56<-11\), and the shaded region is below or on the line? Wait, the line \(7x - 8y=-11\) can be rewritten as \(y=\frac{7x + 11}{8}\). The slope is positive. The intersection point is \((3,4)\). Let's test the inequality with the point \((3,4)\): \(7(3)-8(4)=21 - 32=-11\). Now, take a point in the shaded region, say \((6,0)\) (wait, no, \((6,0)\) is on \(x + y = 6\), not in the shaded region. Take \((0,8)\): \(7(0)-8(8)=-64\). We check the inequality. The line is solid (since the solution includes the line), so the inequality is either \(\leq\) or \(\geq\). Since \(-64\leq - 11\), and the shaded region is such that \(7x-8y\leq - 11\) (because when we plug in a point in the shaded region, the left - hand side of \(7x - 8y\) is less than or equal to \(-11\)). Also, we can see from the graph that the line is solid, so the inequality is \(7x - 8y\leq - 11\).

Answer:

A. \(7x - 8y\leq - 11\)