Problem 36 - Beginning and Intermediate Algebra

Question

Graph using either the test point or slope-intercept method. \(y \leq 4\)

Step-by-Step Solution

Answer

Graph the equation \(y = 4\) as a solid horizontal line passing through the point (0, 4). Choose the test point (0, 0) and check the inequality \(0 \leq 4\), which holds true. Therefore, shade the region below the line, including the line itself. The graph with the shaded region represents the inequality \(y \leq 4\).

Step 1: Graph the corresponding equation

Start by sketching the equation \(y = 4\) on coordinate plane. This is a straight, horizontal line that goes through the point (0, 4) as it passes the point in the y-axis where y = 4. Sketch the line with a solid line since the inequality symbol has an "equal" part (\(\leq\)).

Step 2: Choose a test point

Now, choose a test point that will help us determine which side of the line to shade. The origin (0, 0) is a good test point as long as our line is not passing through it. It is not in this case, so we can use (0, 0).

Step 3: Test the inequality

Plug the test point (0, 0) into the inequality to see if it holds true: \[0 \leq 4\] Since 0 is indeed less than or equal to 4, the inequality holds true for the test point.

Step 4: Shade the appropriate region

Since the test point holds true for the inequality, we shade the side of the line that includes the origin (0, 0). This means shading the region below the line \(y = 4\), including the line since we are dealing with a less than or equal to inequality. The graph with the shaded region represents the inequality \(y \leq 4\).

Key Concepts

Slope-Intercept Method

The slope-intercept method is a fundamental concept used to graph linear equations easily. It helps us to understand and plot the equation in the form of \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. To graph an equation using this method, identify \(b\), plot it on the y-axis, and then use the slope \(m\) to find another point on the line by rising or falling vertically over a specified horizontal distance.
For inequalities like \(y \leq 4\), the process is simplified because the equation \(y = 4\) is already in the slope-intercept form, with no slope (i.e., \(m = 0\)). This makes it a horizontal line. It's important to use a solid line for \(y \leq 4\), indicating that points on the line itself satisfy the inequality.

Test Point Method

The test point method is useful in graphing inequalities, especially when determining which side of the boundary line to shade. This involves picking a point in the coordinate plane that is not on the line to check if it satisfies the inequality.
A common choice for the test point is the origin (0, 0), because substituting \(x = 0\) and \(y = 0\) into any equation is straightforward and quick. For example, in the inequality \(y \leq 4\), substitute (0, 0):
- \[0 \leq 4\]
Since this statement is true, it confirms that the region including (0, 0) is part of the solution set. Thus, you shade the region on the side of the line that contains the test point.

Coordinate Plane

A coordinate plane consists of two perpendicular number lines called axes: the horizontal x-axis and the vertical y-axis. They intersect at a point called the origin (0, 0). The plane is divided into four quadrants that aid in graphing and analyzing mathematical equations or inequalities.
In this exercise, plotting the line \(y = 4\) involves finding the y-coordinate 4 on the y-axis and drawing a horizontal line. The coordinate plane allows us to visualize the inequality \(y \leq 4\) by shading the appropriate side.

• The x-axis can be used as a reference for plotting points horizontally.
• The y-axis is used to plot points vertically and also determines the y-intercepts of the lines.

By understanding the coordinate plane's structure, graphing becomes more intuitive and helps in solving inequalities.

Inequality Graphing

Graphing inequalities is a crucial skill in understanding mathematical relationships visually. Unlike equations that define a line precisely, inequalities describe a region of the coordinate plane.
For graphing \(y \leq 4\), begin by drawing the boundary line \(y = 4\), then determine where to shade. The type of line used matters: a solid line indicates that points on it satisfy the inequality, while a dashed line indicates they do not. Since this inequality includes "\(\leq\)," use a solid line.
Once the line is drawn, use methods like the test point method to decide which side to shade. Shade below the line for \(y \leq 4\) as it includes y-values less than or equal to 4. This shaded area represents all possible solutions to the inequality.

• Solid lines show inclusive inequalities (\(-\infty\) to a solid line point).
• Shading indicates where solutions to the inequality lie in relation to the line.

Understanding inequality graphing allows you to solve, visualize, and interpret mathematical problems effectively.