Problem 79 - A Graphical Approach to College Algebra

Question

Find a system of linear inequalities for which the graph is the region in the first quadrant between and inclusive of the pair of lines \(x+2 y-8=0\) and \(x+2 y=12\)

Step-by-Step Solution

Answer

\(x + 2y \geq 8\), \(x + 2y \leq 12\), \(x \geq 0\), \(y \geq 0\).

Step 1: Understanding the Lines

First, we identify the two given lines, which are described by the equations: \(x + 2y = 8\) and \(x + 2y = 12\). These lines have the same slope (because their coefficients of \(x\) and \(y\) are identical), so they are parallel.

Step 2: Rearranging the Lines

Next, rearrange the lines to the slope-intercept form to better understand the region between them. The equation of the first line is \(y = -\frac{1}{2}x + 4\) and the second line is \(y = -\frac{1}{2}x + 6\).

Step 3: Identifying the Region

The goal is to find the region in the first quadrant where: \(y\) is greater than or equal to \(-\frac{1}{2}x + 4\) (below the first line) and \(y\) is less than or equal to \(-\frac{1}{2}x + 6\) (above the second line). Hence, the system of inequalities based on \(x\) and \(y\) is derived.

Step 4: Creating the Inequalities

The system of inequalities is as follows: \[ x + 2y \geq 8 \] and \[ x + 2y \leq 12 \]. Additionally, to restrict the region to the first quadrant, \(x \geq 0\) and \(y \geq 0\) must also be included.

Key Concepts

First Quadrant

The first quadrant is a crucial concept when dealing with linear inequalities that define regions on a graph. In a 2D coordinate system, the first quadrant is the section where both coordinates, x and y, are positive. This particular quadrant helps simplify many mathematical problems, since it often limits the possible solutions.

When looking at systems of inequalities, being restricted to the first quadrant means we only consider solutions where x and y are both greater than or equal to zero. For the problem at hand, this means adding the inequalities:

• \( x \geq 0 \)
• \( y \geq 0 \)

These additional conditions ensure the solution remains within the first quadrant, thus focusing on feasible values that cater to many real-world problems.

System of Inequalities

A system of inequalities involves solving multiple inequality equations at once. This system creates a region of solutions on a graph rather than a single line or point, and it's useful when trying to define specific areas based on given conditions.

In our example, our two key inequalities are:

• \( x + 2y \geq 8 \)
• \( x + 2y \leq 12 \)

These inequalities describe regions defined by two lines that are parallel. The solution lies between these lines. When graphed together with the first quadrant restriction (\( x \geq 0 \) and \( y \geq 0 \)), the resulting area is the region of interest.

Understanding systems of inequalities helps us frame and solve complex problems, representing areas where certain conditions are met. When considering such systems, it is important to not only take into account the position and relationship of the lines but also ensure any constraints, like first quadrant restrictions, are also met.

Parallel Lines

Parallel lines are two or more lines in a plane that never meet. They are always the same distance apart no matter how far they are extended. In mathematical terms, parallel lines have the same slope. This trait plays an important role in solving linear inequalities.

For the given problem, we have two parallel lines:

• \( x + 2y = 8 \)
• \( x + 2y = 12 \)

Both equations have the form \(y = -\frac{1}{2}x + b\), indicating they have an identical slope of -\(\frac{1}{2}\). The difference in the y-intercept allows them to form two separate yet parallel lines, which enclose a specific region. This region is critical to determining the solution to the system of inequalities.

Recognizing parallel lines helps identify boundaries in a system of inequalities, facilitating the definition of the solution region, especially when further constraints like being in the first quadrant are applied.