Chapter 7: Problem 7

Draw an \(x-y\) coordinate frame and shade the region for which \(x<3\) and \(y>-2\).

Short Answer

Expert verified

Answer: The region where both inequalities hold true is above the horizontal line at \(y = -2\) and to the left of the vertical line at \(x = 3\).

Step by step solution

Understand the inequalities and the coordinate system

We are given two inequalities: \(x<3\) and \(y>-2\). These represent regions in the \(x-y\) coordinate plane where the inequalities hold true. Our task is to graph and shade the region where both inequalities are satisfied simultaneously.

Draw the coordinate system

Draw an \(x-y\) coordinate frame, with x-axis and y-axis intersecting at the origin (0,0). Make sure to label the axes. You may also want to label some points along the axes for easier visualization.

Graph the first inequality

To graph the inequality \(x<3\), first draw a vertical line representing \(x=3\). This line passes through the point (3,0) on the x-axis. All points to the left of this line have x-values less than 3, so the region covered by the inequality \(x<3\) is the entire area to the left of the vertical line at \(x=3\).

Graph the second inequality

To graph the inequality \(y>-2\), first draw a horizontal line representing \(y=-2\). This line passes through the point (0,-2) on the y-axis. All points above this line have a y-value greater than -2, so the region covered by the inequality \(y>-2\) is the entire area above the horizontal line at \(y=-2\).

Shade the intersection of the regions

To find the region where both inequalities are true simultaneously, look for the intersection of the regions represented by each inequality. This intersection lies above the horizontal line at \(y=-2\) and to the left of the vertical line at \(x=3\). Shade this area to represent the solution region.

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Draw an \(x-y\) coordinate frame and shade the region for which \(x<3\) and \(y>-2\).

Short Answer

Step by step solution

Understand the inequalities and the coordinate system

Draw the coordinate system

Graph the first inequality

Graph the second inequality

Shade the intersection of the regions

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