Problem 44 - College Algebra

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&x^{2}+y^{2} \leq 4\\\&x+y>1\end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system is the section of the circle that is above the line, excluding the points on the line.

Step 1: Graph the circle

The first inequality can be rewritten as \(x^2 + y^2 \leq 4\), which represents a circle centered at the origin with a radius of 2. Draw the circle on the plane.

Step 2: Shade the circle

Since the inequality is \(x^2 + y^2 \leq 4\), the solution includes points inside the circle as well as the points on the circle itself. Shade the circle and the area within it.

Step 3: Graph the line from the second inequality

The second inequality can be rewritten as \(y > -x + 1\). This represents a line with a slope of -1 and a y-intercept at 1. Draw this line on the plane.

Step 4: Choose and shade the correct half-plane for the linear inequality

This inequality is \(y > -x + 1\), which means that the solution includes points above this line, not the points on the line. Shade all points above this line.

Step 5: Find the intersection of the two shaded regions

The solution to the system of inequalities is the intersection of the two shaded regions from the previous steps.

Key Concepts

Graphing Inequalities

When graphing inequalities, the aim is to find all solutions that satisfy each inequality in a given system. The graph of an inequality represents a region on the coordinate plane where all points are solutions to the inequality.
For example, if you have an inequality like \(x^2 + y^2 \leq 4\), it describes a region on the graph that is at or within a certain boundary.

• Understand the Boundary: This refers to the line or curve that represents the equality part. In \(x^2 + y^2 = 4\), this would be a circle.
• Shading: For inequalities like \(\leq\) or \( \geq \), you include the boundary in the solution (solid line). For \(<\) or \(>\), do not include it (dashed line).
• Check with Points: Substitute a point into the inequality to see if it's true to know which side to shade.

This method visually represents where solutions to the inequalities lie.

Circle Inequality

A circle inequality like \(x^2 + y^2 \leq 4\) involves all points inside and on the circle. This circle's equation represents a set of points a certain distance from a center point, forming a perfect shape on a graph.

• Center and Radius: Here, the circle has its center at \((0, 0)\) and a radius of 2 because \(\sqrt{4} = 2\).
• Graphing the Circle: Draw a solid circle with radius 2 centered at the origin. Use solid lines because the points on the circle are included in \(\leq\).
• Shading Inside: Since the inequality is less than or equal to, shade the entire area within this circle to show all included solutions.

By understanding circle inequalities, you can easily identify and represent the solutions.

Linear Inequality

Linear inequalities, such as \(y > -x + 1\), describe a half-plane above or below a certain line. Each side of the line contains points that either satisfy or don't satisfy the inequality.

• Slope and Intercept: The line \(y = -x + 1\) has a slope of -1 and y-intercept of 1. It means that as x increases, y decreases.
• Plotting the Line: Start at 1 on the y-axis and go down one unit for every unit right since the slope is -1. This forms the line.
• Selecting the Region: For \(y > -x + 1\), shade above the line to represent all satisfying solutions. Do not include the line itself since it's a strict inequality.

Knowing how to interpret and plot linear inequalities helps in finding the intersection of multiple inequalities.

Solution Set Graphing

Solution set graphing involves finding a common area where multiple inequalities overlap. This region represents all ordered pairs that solve each inequality in the system.

• Intersecting Areas: Look for where the shaded areas from each inequality overlap. This common region is your solution set.
• Checking Boundaries: Ensure understanding of whether boundaries are included based on inequality types.
• Visual Clarity: Use different colors or shading intensities to help distinguish the overlapping areas for clarity.

Graphing solution sets gives a visual way to see all possible solutions and helps solve complex systems through intersection.