Problem 36 - Intermediate Algebra

Question

Graph the solution set of each system of inequalities on a rectangular coordinate system. $$y<1 \text { or } y \leq-5$$

Step-by-Step Solution

Answer

Shade below the line \(y = 1\) and below \(y = -5\).

Step 1: Understanding the Inequalities

We are given two inequalities: \(y < 1\) and \(y \leq -5\). This is a system of inequalities that we need to graph on a coordinate plane.

Step 2: Graphing the First Inequality

To graph \(y < 1\), draw a horizontal dashed line at \(y = 1\) on the Cartesian plane. The region below this line is shaded to indicate all points where \(y\) is less than 1.

Step 3: Graphing the Second Inequality

For \(y \leq -5\), draw a horizontal solid line at \(y = -5\). The region below this line is shaded, indicating all points where \(y\) is less than or equal to -5.

Step 4: Combining the Graphs

Since the inequalities are connected by 'or', the solution set includes any region that satisfies either \(y < 1\) or \(y \leq -5\). Therefore, the entire regions both below \(y = 1\) and below \(y = -5\) are shaded.

Key Concepts

Graphing Inequalities

Graphing inequalities is a method of visually representing all the solutions for an inequality on a graph. Unlike equations, which have exact answers (points), inequalities like \(y < 1\) and \(y \leq -5\) have a range of solutions. This is why graphing can be so useful, as it helps you see the entire set of possible solutions at once.
When you graph an inequality:

• Use a dashed line for "less than" \((<)\) or "greater than" \((>)\) inequalities, indicating that the points on the line are not included.


• Use a solid line for "less than or equal to" \((\leq)\) or "greater than or equal to" \((\geq)\) inequalities, indicating the line itself is part of the solution set.

Once you draw the appropriate line style based on your inequality, shade the correct region of the graph. For example, if you have \(y < 1\), shade the area below the line since \(y\) can be any value less than 1. Similarly, for \(y \leq -5\), shade below it because \(y\) includes -5 and values less than -5. This shading visually emphasizes the multitude of solutions an inequality holds.

Coordinate Plane

The coordinate plane, or Cartesian plane, is a two-dimensional surface formed by two number lines that intersect at a right angle. These number lines are called axes.
An understanding of how the coordinate plane operates is essential for graphing inequalities:

• The horizontal line is known as the x-axis.


• The vertical line is known as the y-axis.


• They intersect at a point called the origin, labeled as \((0, 0)\).

Coordinates are written as pairs: \((x, y)\), where \(x\) is the value on the x-axis, and \(y\) is the value on the y-axis. Each point on the coordinate plane represents this pair of numbers.
When graphing inequalities like \(y < 1\) or \(y \leq -5\), you focus primarily on the y-axis, as these inequalities only involve y. The resulting graph will appear as a horizontal line at the specified y-values.
Graphing on a coordinate plane allows you to easily visualize and solve more complex systems involving multiple inequalities by clearly showing where their solution sets overlap or differ.

Solution Set

A solution set in the context of inequalities refers to all possible values that satisfy the given inequalities. When dealing with systems of inequalities like \(y < 1\) or \(y \leq -5\), the solution set consists of all points on the coordinate plane where the inequalities' conditions hold true.
For example, the solution set for the system given is anything below the line \(y = 1\) or below the line \(y = -5\). In this context, the word "or" is crucial, as it means that points satisfying either one of the inequalities are included in the solution set. This ultimately shapes a large, combined shaded region rather than a single area when graphed.
Understanding the solution set helps in determining feasible regions for various scenarios modeled by inequalities. It also aids in problem-solving, especially when one needs to find common solutions that satisfy multiple constraints. Visual representation through graphs provides an intuitive grasp of what these solution sets look like.