Problem 3 - A Graphical Approach to College Algebra

Question

Graph each inequality. Do not use a calculator. $$x+2 y \leq 6$$

Step-by-Step Solution

Answer

Graph the line through (6,0) and (0,3). Shade below it, including the line.

Step 1: Identify the Type of Inequality

The given inequality is linear: \(x + 2y \leq 6\). We need to rearrange it to represent a line which separates the "less than or equal to" area from others.

Step 2: Convert Inequality to Equation

Convert the inequality to its equivalent equation \(x + 2y = 6\). This will help us determine the boundary line for the inequality.

Step 3: Find Intercepts

Set \(y = 0\) to find the x-intercept: \(x + 2(0) = 6\) gives \(x = 6\). Set \(x = 0\) to find the y-intercept: \(0 + 2y = 6\) gives \(y = 3\). Now, the intercepts are (6,0) and (0,3).

Step 4: Draw the Boundary Line

Plot the points (6,0) and (0,3) on a graph. Draw a solid line through these points, as the inequality includes "equal to." This line represents all points \(x + 2y = 6\).

Step 5: Test a Point

Choose a test point not on the line to determine which side satisfies the inequality. A common choice is the origin (0,0). Substitute into the inequality: \(0 + 2(0) \leq 6\) or \(0 \leq 6\). This is true, so the area below and on the line is the solution.

Step 6: Shade the Solution Region

Shade the half-plane that includes the origin and lies beneath the line. This shaded area represents the solution set \(x + 2y \leq 6\). Any point in this area satisfies the inequality.

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions involving a linear function and an inequality symbol, such as \( \leq \), \( \geq \), \( < \), or \( > \). They show a range of possible solutions, unlike linear equations which have a specific solution. To understand linear inequalities, it is helpful to first understand linear equations, since both involve terms that result in a straight line when graphed.

In the inequality \( x + 2y \leq 6 \), we see that it involves variables \(x\) and \(y\), and it is linear because both variables are to the first power. The inequality symbol \(\leq\) indicates that the inequality includes all points on the boundary line defined by the equivalent linear equation \(x + 2y = 6\), as well as those in a particular half-plane.

Understanding linear inequalities helps us explore the area of possible solutions for a given constraint, which is valuable in fields like optimization and economics.

Coordinate System

A coordinate system is a framework that allows us to pinpoint the location of points on a plane using two numbers, known as coordinates. The most common system is the Cartesian coordinate system, which uses horizontal and vertical axes that divide the plane into quadrants.

In this system, every point is represented as \((x, y)\), where \(x\) is the distance from the vertical y-axis, and \(y\) is the distance from the horizontal x-axis. For the inequality \(x+2y\leq6\), plotting this on a coordinate grid is the first step towards visualization.

• Plot points like the intercepts, which are easy to find and provide guidance on where to draw the boundary line.
• Make sure the axes are marked and scaled appropriately to visualize all necessary points clearly.

By understanding the coordinate system, students can easily navigate through plotting and interpreting graphs of equations and inequalities.

Boundary Line

The boundary line for an inequality is the line that represents the equation you get when you replace the inequality symbol with an equals sign. For the inequality \(x + 2y \leq 6\), the boundary line is found by solving \(x + 2y = 6\).

This line helps separate the coordinate plane into a solution region and a non-solution region. In drawing this line:

• Find the intercepts, which are the simplest points to calculate and plot.
• Use these points to draw the line accurately. If the inequality is \(\leq\) or \(\geq\), draw a solid line to indicate points on the line are included in the solution.

The boundary line is crucial as it defines the limit of one side of the solution region. Mastery of plotting boundary lines results in clear and accurate graphs of inequalities.

Solution Region

The solution region is an area on the graph where all points satisfy the inequality. In our example, the inequality \(x + 2y \leq 6\) creates a solution region that includes the boundary line as well as all points below it.

To locate this region accurately, we use a test point not on the line, like the origin \((0, 0)\). By substituting \((0, 0)\) into the inequality, we find \(0 \leq 6\), which is true. Therefore, the region including \((0, 0)\) is the solution region, and we shade it on the graph to represent this logically.

• Make sure to choose a simple test point, like \((0, 0)\), if possible, for an easy check.
• Once verified, shade that side of the line to clearly illustrate the set of solutions.

This shaded area visually represents where all conditions of the inequality are met, making it a vital part of graphing linear inequalities.