Question

graph the inequality on the axes below.
$-x + y \geq 2$

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(-x + y\geq2\). To get it in the form \(y = mx + b\) (slope - intercept form), we add \(x\) to both sides of the inequality.
\(y\geq x + 2\)

Step2: Identify the boundary line

The boundary line for the inequality \(y\geq x + 2\) is the line \(y=x + 2\). Since the inequality is \(\geq\), the boundary line will be a solid line (because the points on the line are included in the solution set).

• The slope \(m\) of the line \(y=x + 2\) is \(1\) (the coefficient of \(x\)) and the \(y\) - intercept \(b\) is \(2\) (the constant term). So the line passes through the point \((0,2)\) (when \(x = 0\), \(y=2\)). To find another point, we can use the slope. Since the slope is \(1=\frac{1}{1}\), from the point \((0,2)\), we can move \(1\) unit up and \(1\) unit to the right to get the point \((1,3)\), or \(1\) unit down and \(1\) unit to the left to get the point \((- 1,1)\).

Step3: Determine the region to shade

We test a point that is not on the line. A common test point is the origin \((0,0)\). Substitute \(x = 0\) and \(y = 0\) into the inequality \(y\geq x + 2\):
\(0\geq0 + 2\), which simplifies to \(0\geq2\). This is a false statement. So the origin is not in the solution set. Therefore, we shade the region above the line \(y=x + 2\) (because the test point \((0,0)\) which is below the line does not satisfy the inequality, so the solution set is the region that does not contain the origin, i.e., above the line).

To graph the inequality:

1. Draw the solid line \(y = x+2\) (passing through \((0,2)\) and \((1,3)\) etc.).
2. Shade the region above the line \(y=x + 2\).

Answer:

The graph consists of a solid line \(y = x + 2\) (with slope \(1\) and \(y\) - intercept \(2\)) and the region above this line is shaded.