Problem 28 - Algebra 1: Concepts and Skills

Question

Plot the points and draw line segments connecting the points to create the polygon. Then write a system of linear inequalities that defines the polygonal region. Rectangle: \((1,1),(7,1),(7,6),(1,6)\)

Step-by-Step Solution

Answer

The system of linear inequalities that defines the polygonal region is: \(x \geq 1\), \(x \leq 7\), \(y \geq 1\), \(y \leq 6\).

Step 1: Plotting the Points

Firstly, plot the points \((1,1), (7,1), (7,6), (1,6)\) on a Cartesian coordinate system. These points represent the vertexes of the polygon (rectangle) which needs to be created.

Step 2: Drawing the Rectangle

After the points have been plotted, connect them with line segments in the following order: starting at point \((1,1)\) proceed to \((7,1)\), then to \((7,6)\), then to \((1,6)\), and finally back to \((1,1)\). It is essential to ensure the lines are straight and meet at the points to form right angles, as it is a rectangle.

Step 3: Define Linear Inequalities

Now, convert these line segments into linear inequalities. Since the region defined by the rectangle is included, use the 'less than or equal to' and 'greater than or equal to' operators. The top border of the rectangle is represented by \(y \leq 6\), the bottom border by \(y \geq 1\), the right border by \(x \leq 7\) and the left border by \(x \geq 1\). So, the system of linear inequalities defining the region of the rectangle is \[x \geq 1\], \[x \leq 7\], \[y \geq 1\] and \[y \leq 6\].

Key Concepts

Plotting Points

Plotting points is the first step in creating any shape in a coordinate system. Think of the plane as a map, and points as specific locations on this map. Each point is designated by a pair of numbers known as coordinates. The first number indicates the position along the horizontal axis (x-axis) and the second along the vertical axis (y-axis).
For instance, the point (1,1) would lie one unit right and one unit up from the origin (where the x-axis and y-axis meet). To effectively plot a point, follow these steps:

• Start from the origin, which is at (0,0).
• Move along the x-axis to the position indicated by the first number.
• Then move parallel to the y-axis to reach the height of the second number.
• Mark this spot with a small dot to represent the plotted point.

After plotting all the points needed, you can connect them to visualize the shape, which begins the process of forming geometrical figures like rectangles or triangles on a grid.

Linear Inequalities

Once you've plotted the points and formed a shape, you can define the space it occupies with linear inequalities. Linear inequalities express the concept of a boundary on a plane that divides it into regions. They help define which part of the space is included in the region you're interested in.
These inequalities are derived from the line equations that form each boundary of the shape. For instance, if you have a line that runs vertically along the x-axis at 1, this can be expressed as the inequality \(x \geq 1\), indicating all points to the right of this line fall within your area of interest. Similarly, a horizontal line might be expressed as \(y \leq 6\), showing that everything below this boundary is included.
To establish a system of linear inequalities that defines a rectangular region:

• Identify each side of the rectangle as a line.
• Translate these lines into inequalities. Use '\(\leq\)' or '\(\geq\)' when the border is inclusive, meaning it defines part of the outer edge of the region.

For a rectangle, expect your system to include exactly four inequalities, each representing one boundary of the shape.

Coordinate System

A coordinate system is a plane divided by two lines, known as axes, which intersect at a right angle at the origin. This system, called the Cartesian coordinate system, helps in determining the exact position of each point by using two numbers referred to as coordinates. These coordinates dictate movement on this grid. The horizontal number line is the x-axis, while the vertical number line is the y-axis. Each point or location on this plane can be precisely located using coordinates (x, y), where 'x' represents horizontal positioning, and 'y' shows vertical movement.
You'll see how the axes divide the plane into four quadrants:

• First Quadrant: where both x and y are positive.
• Second Quadrant: where x is negative, but y is positive.
• Third Quadrant: where both x and y are negative.
• Fourth Quadrant: where x is positive, but y is negative.

This grid system allows for easy visualization and plotting of mathematical concepts, such as shapes, points, and lines. Understanding how to navigate this system is essential for accurate plotting and interpretation of data when working with systems of equations or inequalities.