Problem 29 - 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. Triangle: \((0,0),(-7,0),(-3,5)\)

Step-by-Step Solution

Answer

The points are plotted and then connected to form the triangle. The system of linear inequalities defining the polygonal region of the triangle is \(y\geq0\), \(y\geq-\frac{5}{3}x\), and \(x\geq-7\)

Step 1: Plotting the Points

Plot the points provided which are \((0,0),(-7,0),(-3,5)\) on the coordinate plane. The first number in the pair is the x-coordinate which aligns with the horizontal axis and the second number in the pair is the y-coordinate which aligns with the vertical axis.

Step 2: Drawing the Line Segments

Connect these points using straight line segments to form a triangle. The line segment connecting the points \((0,0),(-7,0)\) is along x-axis and it's equation is \(y = 0\). The line segment connecting the points \((0,0),(-3,5)\) can be represented bythe equation \(y = -\frac{5}{3}x\). Vertical line through -7 is \(x = -7\).

Step 3: Defining the Polygonal Region with Inequalities

The polygonal region of the triangle is defined by the following inequalities \(y\geq0\), \(y\geq-\frac{5}{3}x\), and \(x\geq-7\)

Key Concepts

Plotting Points

Understanding how to plot points is fundamental in graphing a system of linear inequalities. Start with a clean coordinate plane, which consists of a horizontal axis (the x-axis) and a vertical axis (the y-axis) that intersect at the origin, marked as \(0,0\). To plot a point, such as \(0,0\), you simply find where the x-coordinate and y-coordinate meet on their respective axes. This particular point is at the origin where both values are zero.

For \( -7,0 \) first move 7 units to the left from the origin (because the x-value is negative) and since the y-value is 0, you stay at the x-axis. For the point \( -3,5 \) move 3 units left (negative direction on the x-axis) and then 5 units up (positive direction on the y-axis). Ensuring proper placement of these points is crucial for the next steps.

Line Segments

Once you've plotted the points on the coordinate plane, drawing line segments between those points is the next task. A line segment is a part of a line that is bounded by two distinct end points that you've plotted. To correctly draw a segment, use a ruler to connect the dots straightly.

For our exercise, you will connect the point \(0,0\) to \( -7, 0 \), which will lie on the x-axis since the y-value does not change, creating a horizontal line segment. Similarly, connecting \(0,0\) to \( -3, 5 \) will give you a line that rises from the x-axis at a defined slope. Line segments are the building blocks of our polygonal shape and they must be accurate to define the correct region later on.

Coordinate Plane

The coordinate plane is the 'stage' for graphing equations and inequalities. It consists of two number lines that are perpendicular to each other: the horizontal x-axis and the vertical y-axis. Concatenating line segments on this plane helps visualize mathematical relationships and solutions.

In the context of our exercise, after plotting points and connecting them with line segments, you actually draw the triangle on the coordinate plane. This visual aid is essential for interpreting and analyzing geometric shapes and algebraic conditions, especially when they get complex. Learning to navigate the coordinate plane with confidence is key for any student tackling geometry or algebra.

Polygonal Region

A polygonal region is a closed geometric figure comprised of straight lines, which in this scenario, is a triangle. When this region is within a coordinate system, it can be represented not just visually but also algebraically with inequalities that define the set of points inside the polygon.

In our exercise, the inequalities describing the polygonal region are derived from the lines forming the polygon's boundaries. These inequalities, \(y \geq 0\), \(y \geq -\frac{5}{3}x\), and \(x \geq -7\) represent the region enclosed by the line segments, which include all the points that satisfy all of these conditions simultaneously. It's like drawing an invisible fence where points inside respect the rules given by the inequalities.