Problem 35 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x \leq 2 \\ y \geq-1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set of the system of inequalities is the upper left quadrant formed by the lines x=2 and y=-1, including the lines x=2 and y=-1.

Step 1: Graph the Inequality x ≤ 2

Begin by drawing the vertical line x=2 on a coordinate plane. Since the inequality is less than or equal to, the corresponding area is on or to the left of the line x=2.

Step 2: Graph the Inequality y ≥ -1

Now draw the horizontal line y=-1 on the same coordinate plane. Since the inequality is greater than or equal to, this means that the appropriate section is on or above the line y=-1.

Step 3: Determine the Intersection of the Two Inequalities

The area that satisfies both conditions (x ≤ 2 and y ≥ -1) is on or to the left of the line x=2 and on or above the line y=-1. Therefore, the intersection area of these two inequalities is the upper left quadrant formed by these two lines.

Key Concepts

Graphing Inequalities

Graphing inequalities on a coordinate plane is a fundamental skill in mathematics. It involves visualizing the solution set of inequalities by shading the area that satisfies each inequality on the grid. To start, one must understand the inequality sign:

• "≤" or "≥" means that the line is included in the solution, so it is drawn as a solid line.
• "<" or ">" means the line is not included in the solution, so it is drawn as a dashed line.

For example, if you have the inequality \(x \leq 2\), you draw a solid vertical line at \(x = 2\) and shade everything to its left. Similarly, for \(y \geq -1\), a solid horizontal line is drawn at \(y = -1\), and the region above it is shaded. This way, graphing helps to visually identify the areas that satisfy the inequality conditions.

Intersection of Inequalities

The intersection of inequalities represents the region where all given inequalities are true at the same time. This concept is crucial when solving a system of inequalities, as it highlights the common solution area on the graph.
In the exercise example, the system involves:

• \(x \leq 2\)
• \(y \geq -1\)

The intersecting area, the solution set, can be identified by finding where the shaded regions from each inequality overlap. This overlapping area satisfies both conditions simultaneously. It appears in the upper left quadrant formed by the intersection of the vertical line \(x = 2\) and the horizontal line \(y = -1\). Thus, the solutions to the system of inequalities lie within this region.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where each point is determined by an ordered pair of numbers known as coordinates. The plane is divided into four quadrants by the horizontal x-axis and the vertical y-axis.

• The positive x-axis extends to the right, while the negative extends to the left.
• The positive y-axis extends upwards, while the negative extends downwards.

When graphing systems of inequalities, the coordinate plane becomes a powerful tool to visualize and solve inequalities. By plotting the lines of the individual inequalities, we can effectively utilize the coordinate plane to identify and shade the solution region. Each quadrant of the plane provides a unique way to interpret the positioning and relationship of these inequalities, making it easy to see how they interact and where their solutions lie.