Graph the following first-degree inequality in two unknowns.
$$ y > {\color{red}2} x - {\color{red}2} $$
To graph the inequality \(y > 2x - 2\), we can follow these steps:
1. Start by graphing the line \(y = 2x - 2\).
2. Since the inequality is \(y > 2x - 2\), we need to determine whether the region above or below the line is shaded.
3. Choose a test point not on the line. Let's say we use the point (0, 0).
4. Substitute the values of x and y from the test point into the inequality. If the inequality holds true, shade the region that contains the test point; otherwise, shade the other region.
5. Draw a dashed line to represent the original equation \(y = 2x - 2\), since the inequality does not include the equal sign.
Let's go through these steps:
1. To graph the line \(y = 2x - 2\), we can start by plotting two points on the line. For instance, when x = 0, y will be -2, and when x = 1, y will be 0. Connect these points with a straight line.
2. Since the inequality is \(y > 2x - 2\), we need to determine whether the region above or below the line is shaded. Because the inequality contains a "greater than" sign, the region above the line will be shaded.
3. Let's use the point (0, 0) as our test point.
4. Substitute x = 0 and y = 0 into the inequality: \(0 > 2(0) - 2\). Simplifying, we have \(0 > -2\), which is true. Since the inequality is true when using the test point (0, 0), we shade the region above the line.
5. Draw a dashed line to represent the original equation \(y = 2x - 2\).
Now we have the graph of the inequality \(y > 2x - 2\), where the shaded region is above the dashed line.