Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Graph each inequality. $$y \geq x-2$$

Step-by-Step Solution

Answer

Graph it with a solid line for \(y = x - 2\) and shade above.

Step 1: Recognize the Standard Line Equation

The inequality given is \(y \geq x - 2\). This can be interpreted as the line equation \(y = x - 2\), with a boundary to consider since it is an inequality.

Step 2: Identify the Boundary Line

The line \(y = x - 2\) is the boundary of the inequality. This line has a slope of 1 and a y-intercept of -2. Start by plotting the y-intercept; at point \((0, -2)\) on the Cartesian plane.

Step 3: Plot the Line

Using the slope of 1, from the intercept \((0, -2)\), move up 1 unit and right 1 unit to another point \((1, -1)\). Continue this pattern to draw the straight line accurately. Since the inequality symbol is \(\geq\), the line will be solid, indicating that points on the line satisfy the inequality.

Step 4: Determine the Shaded Region

For \(y \geq x - 2\), shade the area above the line (including the line itself) because \(y\) values greater than \(x-2\) satisfy the inequality. Test a point not on the line, for example, \((0,0)\). Substitute it back: \(0 \geq 0 - 2\), or \(0 \geq -2\) which is true, confirming the shade is correct.

Key Concepts

Cartesian Plane

The Cartesian plane, also known as the coordinate plane, is a two-dimensional surface where we can graph mathematical equations and inequalities. It consists of two perpendicular number lines, one horizontal (x-axis) and one vertical (y-axis), which intersect at a point called the origin, denoted as (0,0). This plane allows mathematicians and students alike to visually interpret algebraic expressions and relationships.

Points on this plane are described using a pair of numbers called coordinates, written as (x, y). The first number represents the position on the horizontal axis, while the second number represents the position on the vertical axis.

• Quadrants: The plane is divided into four sections, referred to as quadrants, each with a distinct combination of positive and negative values for x and y.
• Plotting Points: To plot a point, locate the x-coordinate on the x-axis, then find the y-coordinate on the y-axis, and mark where these intersect.

Understanding the Cartesian plane is essential for graphing inequalities because it provides a structured environment to visualize solutions.

Linear Equations

Linear equations form straight lines when graphed on the Cartesian plane. They are fundamental in understanding graphing, as they represent a constant relationship between x and y. The general format of a linear equation is \(y = mx + b\), where m is the slope and b is the y-intercept.

To graph a linear equation, you need to identify:

• Slope: This tells us how steep the line is, or how much y changes for a change in x. In the equation \(y = x-2\), the slope m is 1, indicating a gentle slope rising diagonally across the plane.
• Y-intercept: This is the starting point on the y-axis. For our equation, it is -2, indicated by the point (0,-2).

These parts of the equation guide us in plotting accurate graphs for solving questions involving inequalities. Recognizing these features accurately is key to mastering the graphing of linear equations.

Slope-Intercept Form

The slope-intercept form is a simple way of writing the equation of a line, making it easy to graph. It is given by the formula \(y = mx + b\), where m is the slope and b is the y-intercept.

This form is particularly useful because it provides direct insights into the graph's characteristics:

• Slope (\(m\)): Indicates the direction and steepness of the line. A positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards.
• Y-intercept (\(b\)): Tells us where the line crosses the y-axis. Using this point, you can start the line's graph before using the slope to extend it across the plane.

For example, in the inequality \(y \geq x - 2\), the line \(y = x - 2\) serves as the boundary. The slope-intercept form helps in quickly plotting this line before shading the appropriate region for inequalities.