Problem 59 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The inequality \(2 x-3 y<6\) contains a "less than" symbol, so its graph lies below the boundary line.

Step-by-Step Solution

Answer

The statement 'The inequality \(2 x-3 y<6\) contains a 'less than' symbol, so its graph lies below the boundary line.' makes sense because the graph for this inequality would indeed lie below the boundary line as the origin, which lies below the line, solves the inequality.

Step 1: Understand the inequality

The inequality given is \(2x - 3y < 6\). The corresponding boundary line is obtained by replacing the '<' with '=' and creating the equation \(2x - 3y = 6\). This line divides the cartesian plane into two regions.

Step 2: Test a point

Take a point from one of the regions, typically the origin (0,0) is taken as it's easier to work with, and substitute it into the inequality. If the inequality holds true, then the point lies in the solution set of the inequality.

Step 3: Examine the validity of the statement

If 'below the boundary line' refers to the region that includes the origin (0,0), then the statement only makes sense if the origin solves the inequality. If the origin does not solve the inequality, then the solution lies above the boundary line, not below, making the statement false.

Step 4: Evaluating the Inequality

Substituting origin (0,0) in the inequality, we get \(2*0 - 3*0 < 6\) which simplifies to \(0 < 6\), a true statement. Hence the solution set lies in the region which includes the origin.

Key Concepts

Cartesian Plane

A cartesian plane, also known as the coordinate plane, is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). The point where these axes intersect is known as the origin, denoted as (0,0). This system allows us to plot points and graph equations on a grid.

• Points on the cartesian plane are denoted as (x, y), where x represents the horizontal position, and y represents the vertical position.
• Equations in two variables can be graphed on this plane, dividing it into different regions.
• Each point on the plane has a unique pair of x and y values, which makes it easy to determine its location.

Understanding the cartesian plane is fundamental to graphing inequalities as it helps visualize which parts of the plane are included in the solution set for the inequality.
For example, the inequality in our exercise splits the plane into two regions, helping determine where the inequality holds.

Boundary Line

The boundary line serves a crucial role when graphing inequalities. It acts as the dividing line on the cartesian plane between the region that satisfies the inequality and the region that does not. The boundary line is derived from the linear equation formed by replacing the inequality symbol with an equal sign. In the given exercise, the boundary line comes from turning the inequality \(2x - 3y < 6\) into the equation \(2x - 3y = 6\). This line helps visually separate solutions of the inequality:

• Solid boundary lines are used for \( \geq \) or \( \leq \) inequalities, indicating the boundary itself is part of the solution set.
• Dashed boundary lines are used for \( > \) or \( < \) inequalities, indicating the boundary is not part of the solution set.

The orientation, whether horizontal or vertical, depends on the coefficients of x and y. In our example, substituting points on one side of the line, such as the origin, helps decide which side of the boundary line is the solution set.

Solution Set

The solution set of an inequality in a two-variable equation consists of all the points in the cartesian plane that satisfy the inequality.
The procedure to find the solution set includes graphing the corresponding boundary line and testing which region satisfies the inequality.For the inequality \(2x - 3y < 6\), once the boundary line \(2x - 3y = 6\) is graphed:

• Choose a test point that is not on the boundary line to determine which side of the line constitutes the solution set. Commonly, the origin (0,0) is the first choice because it simplifies calculations.
• Substitute the test point into the inequality. If the inequality is true for that point, the region containing that point is part of the solution set.

After testing, since substituting (0,0) into \(2x - 3y < 6\) results in a true statement, the region containing this point is indeed the solution set. Remember, the solution set represents all possible (x, y) pairs that satisfy the inequality, offering a visual summary of solutions on the graph.