Question

graph the following inequality.
4x + y > 4
use the graphing tool to graph the inequality.
click to enlarge graph

Explanation:

Step1: Rewrite the inequality in slope - intercept form

We start with the inequality \(4x + y>4\). To get it in the form \(y = mx + b\) (slope - intercept form), we solve for \(y\). Subtract \(4x\) from both sides of the inequality: \(y>- 4x + 4\).

Step2: Graph the boundary line

The boundary line for the inequality \(y>-4x + 4\) is the line \(y=-4x + 4\). Since the inequality is \(>\) (not \(\geq\)), the boundary line should be a dashed line.

• To find the \(y\) - intercept, set \(x = 0\). Then \(y=-4(0)+4=4\), so the line passes through the point \((0,4)\).
• To find the \(x\) - intercept, set \(y = 0\). Then \(0=-4x + 4\), which gives \(4x = 4\) or \(x = 1\), so the line passes through the point \((1,0)\). Plot these two points \((0,4)\) and \((1,0)\) and draw a dashed line through them.

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 \(y>-4x + 4\). We get \(0>-4(0)+4\), which simplifies to \(0 > 4\). This is false. So we shade the region that does not contain the origin. In other words, we shade the region above the line \(y=-4x + 4\) (since the test point \((0,0)\) is below the line and the inequality is not satisfied at \((0,0)\), we shade the opposite side).

(Note: Since this is a graphing problem, the final answer is the graph with the dashed line \(y = - 4x+4\) and the region above the line shaded. If we were to describe the key features: dashed line through \((0,4)\) and \((1,0)\), shading above the line.)

Answer:

The graph consists of a dashed line \(y=-4x + 4\) (passing through \((0,4)\) and \((1,0)\)) with the region above the line shaded.