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 < -x + 4\\
\\y > 2x - 5\\

Explanation:

Graph the first inequality

To graph \(y < -x + 4\), we first find the boundary line \(y = -x + 4\).
Using the Linear Equations Graphing knowledge point, we identify the \(y\)-intercept at \((0, 4)\) and the slope as \(-1\).
Since the inequality is strict (\(<\)), we draw a dashed boundary line.
We test the point \((0,0)\): \(0 < 0 + 4\) is true, so we shade the region below the line.

Graph the second inequality

To graph \(y > 2x - 5\), we find the boundary line \(y = 2x - 5\).
Using the Linear Equations Graphing knowledge point, we identify the \(y\)-intercept at \((0, -5)\) and the slope as \(2\).
Since the inequality is strict (\(>\)), we draw a dashed boundary line.
We test the point \((0,0)\): \(0 > 0 - 5\) is true, so we shade the region above the line.

Find the intersection region

The solution set is the region where both shaded areas overlap.
This is the region below the line \(y = -x + 4\) and above the line \(y = 2x - 5\).

Identify a point in the solution set

We choose a test point within the overlapping region, such as \((0, 0)\).
Let's verify \((0, 0)\):

1. \(0 < -0 + 4 \implies 0 < 4\) (True)
2. \(0 > 2(0) - 5 \implies 0 > -5\) (True)

Thus, \((0, 0)\) is a valid solution.

Answer:

The system of inequalities is solved graphically by plotting the dashed boundary lines \(y = -x + 4\) and \(y = 2x - 5\), then shading the region below the first line and above the second line.

A coordinate point in the solution set is:

$$(0, 0)$$

(Note: Any point in the overlapping shaded region, such as \((1, 1)\), \((2, -1)\), or \((0, -2)\), is also a correct answer.)