Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the inequality. $$2 y-x \geq 4$$

Step-by-Step Solution

Answer

The graph of the inequality \(2y - x \geq 4\) is a straight line passing through the point (0, 2) with slope of \frac{1}{2}. The shading is above the line.

Step 1: Rewrite inequality as an equation

The first step is to rewrite the inequality \(2y - x \geq 4\) as an equation: \(2y - x = 4\). You should do this to make the plotting easier, as the line will separate the graph into regions.

Step 2: Rearrange the equation in terms of y

Rearrange the equation \(2y - x = 4\) in terms of \(y\). To do this, add \(x\) to both sides to get \(2y = x + 4\). Then, divide both sides by 2 to isolate \(y\), resulting in \(y = \frac{x}{2} + 2\). This form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, is easier to graph.

Step 3: Plot the line

Next, plot the equation \(y = \frac{x}{2} + 2\) on a graph. Start by plotting the y-intercept, which is \(2\). Then, use the slope \(\frac{1}{2}\) to find another point on the line. Starting from the y-intercept go 1 unit up and 2 units to the right, mark this point. Then draw a line through these two points.

Step 4: Identify the inequality region

Finally, identify the inequality region. Since the inequality is \(\geq\), this means that the solution includes the line and the area that the line covers. If it was \(>\), it would only include the area above the line. In this case, choose a test point, if it satisfies the inequality shade the region containing the test point, otherwise, shade the other region.