Question

the following graph represents an inequality. select the ordered pairs which are solutions to it. draw (5, -2) (1, 7) (0, 0) (2, -2) (-2, 1) (6, -14)

Explanation:

Step1: Find the equation of the boundary line

The boundary line passes through \((0, 0)\) and \((2, -6)\) (we can find another point by looking at the graph). The slope \(m=\frac{-6 - 0}{2-0}=- 3\). Using the slope - intercept form \(y = mx + b\) (with \(b = 0\) since it passes through the origin), the equation of the boundary line is \(y=-3x\). The line is dashed, so the inequality is either \(y>-3x\) or \(y < - 3x\). The shaded region is to the right of the line (where \(y\) values are greater than \(-3x\) for a given \(x\)), so the inequality is \(y>-3x\).

Step2: Test each ordered pair

• For \((5,-2)\): Substitute \(x = 5\) and \(y=-2\) into \(y>-3x\). \(-2>-3\times5=-15\). Since \(-2>-15\) is true, \((5, - 2)\) is a solution.
• For \((1,7)\): Substitute \(x = 1\) and \(y = 7\) into \(y>-3x\). \(7>-3\times1=-3\). Since \(7>-3\) is true, \((1,7)\) is a solution.
• For \((0,0)\): Substitute \(x = 0\) and \(y = 0\) into \(y>-3x\). \(0>-3\times0 = 0\) is false. So \((0,0)\) is not a solution.
• For \((2,-2)\): Substitute \(x = 2\) and \(y=-2\) into \(y>-3x\). \(-2>-3\times2=-6\). Since \(-2>-6\) is true, \((2,-2)\) is a solution.
• For \((-2,1)\): Substitute \(x=-2\) and \(y = 1\) into \(y>-3x\). \(1>-3\times(-2)=6\) is false. So \((-2,1)\) is not a solution.
• For \((6,-14)\): Substitute \(x = 6\) and \(y=-14\) into \(y>-3x\). \(-14>-3\times6=-18\). Since \(-14>-18\) is true, \((6,-14)\) is a solution.

Answer:

\((5, - 2)\), \((1,7)\), \((2,-2)\), \((6,-14)\)