Problem 31 - Precalculus

Question

In Exercises \(21-50,\) graph each system of inequalities or indicate that the system has no solution. $$\begin{aligned} &\begin{array}{l} y > x \\ x < 0 \end{array}\\\ &y < 4 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is the region below the line \(y = 4\), above the line \(y = x\), and left of the \(y\)-axis.

Step 1: Understand the System of Inequalities

We have a system of three inequalities: \(y > x\), \(x < 0\), and \(y < 4\). Our goal is to graph these inequalities on a coordinate plane to find the region where all of them are satisfied simultaneously.

Step 2: Graph the Inequality \(y > x\)

Start by graphing the line \(y = x\), which is a straight line through the origin (0,0) with a slope of 1. Since the inequality is \(y > x\), shade the region above this line, excluding the line itself.

Step 3: Graph the Inequality \(x < 0\)

Graph the line \(x = 0\), which is the y-axis itself. Since \(x < 0\), shade the region to the left of the y-axis, excluding the y-axis.

Step 4: Graph the Inequality \(y < 4\)

Graph the horizontal line \(y = 4\). Since \(y < 4\), shade the area below this line, excluding the line itself.

Step 5: Determine the Solution Region

Find the region where the shaded areas of all three inequalities overlap. This is the solution region where all inequalities are simultaneously satisfied. Shade this intersection area to indicate the solution of the system.

Key Concepts

Graphing Inequalities

Understanding how to graph inequalities is crucial when dealing with systems of inequalities. To graph an inequality, you first graph its associated equation. This involves finding a line, which provides a boundary for the inequality on the coordinate plane. Once you've graphed the line, you'll decide which side of the line satisfies the inequality.

• For inequalities with ">" or "<", the boundary line is dashed, indicating that points on the line aren't included in the solution.
• For inequalities with "≥" or "≤", the line is solid, showing that points on the line are included.

For example, when graphing the inequality \( y > x \), you plot the line \( y = x \), then shade the entire region above this line. Only the shaded region satisfies the inequality.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates: \( (x, y) \). The x-coordinate represents horizontal movement, while the y-coordinate measures vertical movement. This grid-like structure helps visualize mathematical concepts like equations and inequalities.

• The x-axis is the horizontal line where \( y = 0 \).
• The y-axis is the vertical line where \( x = 0 \).
• These axes divide the plane into four quadrants.

In graphing systems of inequalities, the coordinate plane becomes a canvas where we plot all relevant lines and shaded regions. Each inequality contributes to a particular section of the plane defined by its solutions.

Solution Region

The solution region is the area on the coordinate plane where all the inequalities in a system are true at the same time. This region is found by overlaying the individual shaded areas from each inequality.
When graphing, each inequality carves out a part of the plane by shading everything either above, below, to the right, or to the left of its boundary line. The solution region is identified by looking for where these shaded areas intersect. This common zone is shaded to indicate that it satisfies every inequality in the system.
In our exercise, by considering inequalities like \( y > x \), \( x < 0 \), and \( y < 4 \), we find a triangular region on the left side of the y-axis, below the line \( y = 4 \). This is your solution area which is the focus of the system.

Intersection Area

The intersection area in the context of graphing inequalities refers to the overlap of shaded regions created by individual inequalities. This is where each inequality holds true simultaneously. Finding this region is like solving a puzzle: each piece (inequality) contributes to a part of the picture (solution area).

• Observe the separately shaded regions for each inequality on the graph.
• Identify the common area where the shades from all the inequalities overlap.

This overlapping area is the feasible region that satisfies all conditions set by the system of inequalities. For the exercise given, the intersection is a triangular region in the negative x-axis territory, below \( y = 4 \) and above \( y = x \), neatly illustrating the solution to the inequalities. This overlap is not only the result but also the visual depiction of solving systems of inequalities.