Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the inequality. $$y<2-x$$

Step-by-Step Solution

Answer

The graph describes a region below the line \(y = 2 - x\). The line \(y = 2 - x\) itself is a dashed line, indicating it is not a part of the solution to the inequality.

Step 1: Understanding the inequality

The given inequality is \(y < 2 - x\). It describes y in terms of x and indicates that y is less than 2 minus x.

Step 2: Creating table of values

Next, we create a table of values for \(x\) and \(y\). We can start by setting \(x\) to -2, -1, 0, 1 and 2. Using these values in the inequality \(y < 2 - x\), we get corresponding y-values. Keep in mind that y should be less than these values.

Step 3: Plotting the graph

The graph is plotted by putting these pairs of (x, y) values onto the graph. Since we're dealing with an inequality (and not just an equation), we're interested in sketching a region of the graph, not just a line. Due to the '<' in the inequality, we draw a dashed line for the equation \(y = 2 - x\) to show that we don't include this line in the solution. The solution to the inequality will be the region below this line, because we want to find where \(y\) is less than \(2 - x\). We choose a test point (like the origin, (0,0), unless the line passes through the origin), and evaluate the inequality. If it holds true, then the region containing the test point is the solution region. If it doesn't hold true, then the region not containing the test point is the solution region.

Step 4: Interpreting the graph

The final sketch of the graph should indicate the region where \(y\) is less than \(2 - x\). Notice that this will be all points under the line \(y = 2 - x\).