91Ó°ÊÓ

Before graphing inequalities, it is crucial to understand the coordinate plane, which is the surface on which we plot points, lines, and curves. It is comprised of two perpendicular number lines that intersect at their zero points: the horizontal axis (x-axis) and the vertical axis (y-axis). The point of intersection is known as the origin, labeled as point (0, 0).

The coordinate plane is divided into four quadrants, each representing a unique combination of sign (+ or -) for the x and y values. The first quadrant (top right) has both x and y positive, while the second (top left), third (bottom left), and fourth (bottom right) have differing signs. When graphing a linear inequality like the one in our exercise, you start by plotting points and drawing a line using the two axes. The line divides the plane into two regions and helps to visualize where the inequality holds true.

The slope-intercept form is one of the easiest ways to plot a line on the coordinate plane. It is written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis. In our given exercise, the equation \(y = 3x + 1\) tells us that the slope (rise over run) of the line is 3, indicating that for every 1 unit we move to the right along the x-axis, we move up 3 units. The y-intercept is 1, revealing that the line crosses the y-axis at the point (0, 1).

When graphing inequalities, we still use the slope-intercept form as a starter. However, since the exercise involves an inequality (\(y < 3x + 1\)) rather than an equation, the line we draw will be dotted, signifying that points on the line itself are not included in the solution set.

To visually represent the solution set of a linear inequality, shading is used on the coordinate plane. After plotting the line in slope-intercept form and determining it needs to be dotted to represent 'less than', the next step lies in deciding which side of the line to shade. For our exercise with the inequality \(y < 3x + 1\), we shade below the line because we are considering all y-values that are less than the value of \(3x + 1\) for any x.

In general, if the inequality is 'less than' or 'less than or equal to', we shade below the line. If it is 'greater than' or 'greater than or equal to', we shade above the line. The shading helps differentiate between where the inequality is or is not satisfied, making it a crucial step in graphing linear inequalities.