Question

graph the inequality.
4x - y < -3

Explanation:

Answer:

To graph \(4x - y < -3\), follow these steps:

Step 1: Rewrite the inequality in slope - intercept form (\(y=mx + b\))

Start with the given inequality \(4x-y<-3\).
Subtract \(4x\) from both sides: \(-y<-4x - 3\).
Multiply both sides by \(- 1\). Remember that when we multiply or divide an inequality by a negative number, the direction of the inequality sign changes. So we get \(y > 4x+3\).

Step 2: Graph the boundary line

The boundary line for the inequality \(y>4x + 3\) is the line \(y = 4x+3\). Since the inequality is \(y>4x + 3\) (not \(y\geq4x + 3\)), the boundary line should be a dashed line.

• The slope \(m\) of the line \(y = 4x+3\) is \(4\) (or \(\frac{4}{1}\)) and the \(y\) - intercept \(b\) is \(3\).
• Plot the \(y\) - intercept: The point \((0,3)\) is on the line.
• Use the slope to find another point. From \((0,3)\), move up \(4\) units and then to the right \(1\) unit. This gives the point \((1,7)\). Draw a dashed line through the points \((0,3)\) and \((1,7)\) (and other points you may find using the slope) to represent the boundary line \(y = 4x+3\).

Step 3: Shade the region

To determine which side of the line to shade, we can use a test point. A common test point is \((0,0)\) (as long as it is not on the boundary line).
Substitute \(x = 0\) and \(y = 0\) into the inequality \(y>4x + 3\):
\(0>4(0)+3\)
\(0>3\), which is false. So the point \((0,0)\) is not in the solution region.
We shade the region above the dashed line \(y = 4x+3\) (because the test point \((0,0)\) which is below the line does not satisfy the inequality, so the solution region is above the line).

The final graph will have a dashed line with equation \(y = 4x + 3\) and the region above the line shaded.