Problem 48 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution of the given system of inequalities consists of the intersected shaded region of the circle \(x^2 + y^2 \leq 4\) and the area above the line \(x + y > 1\).

Step 1: Plot the Graph of the First Inequality

The first inequality is \(x^2 + y^2 \leq 4\). This can be graphed as a circle centered at the origin (0,0) with a radius of 2. Because the inequality is \(\leq\), we want to shade in the area within and on the boundary of the circle, as this includes all points (x, y) that satisfy the first inequality.

Step 2: Plot the Graph of the Second Inequality

Next, plot the line of the second inequality \(x + y = 1\). This is a straight line that cuts the plane into two halves. The inequality \(x + y > 1\) signifies that we are interested in the region above the line, not on or below it. So, plot the line and shade the region above it.

Step 3: Identify the Solution Set for Both Inequalities

The solution set of the system of inequalities will be the region in the plane which satisfies both inequalities simultaneously. This region will be represented by the intersection of the shaded areas from Step 1 and Step 2.

Key Concepts

Algebraic Inequalities

When we talk about algebraic inequalities, we're referring to mathematical expressions that show the relationship between two values where one is not strictly equal to the other. Instead, they are related by statements like 'less than' (\textless), 'greater than' (\textgreater), 'less than or equal to' (\textless=), or 'greater than or equal to' (\textgreater=). Unlike equations that have a single solution, inequalities usually have a range of possible solutions.

For example, the inequality \( x^2 + y^2 \textless= 4 \) includes all (x, y) coordinate pairs that, when plugged into this expression, will give a result that is less than or equal to 4. In a graphical representation, this would include not just a line or a point but an entire region on a coordinate plane. Understanding how to solve and graph these inequalities is crucial since it extends to concepts like optimization and probability in higher mathematics.

Plotting Inequalities

When it comes to plotting inequalities, the process often involves transforming the problem into something visual wherein each point in the region represents a potential solution. To plot an inequality, you first graph the 'boundary' – this could be a line (in the case of linear inequalities), a curve, or another shape. Then you determine which side of the boundary to shade to represent the solution set.

For instance, the line \( x + y = 1 \) represents the boundary for the inequality \( x + y > 1 \). The inequality suggests that the solutions lie above the line since they should be greater than the value 1 when x and y are summed. Here, proper shading is key. If the inequality includes the boundary (using \textless= or \textgreater=), the boundary line is drawn solidly and that line itself is part of the solution set. For strict inequalities (using \textless or \textgreater), the boundary line is typically dashed to indicate that it is not included in the set of solutions.

Solution Set of Inequalities

The solution set of inequalities in a system is the set of all points that satisfy all the inequalities at once. In other words, it's the intersection of all individual solution sets from each inequality in the system. To find this on a graph, you'd look for the region where the shaded areas of all inequalities overlap.

Determining the solution set visually can be straightforward but may require careful consideration, especially with complex systems. When given a system like \( x^{2}+y^{2} \textless= 4 \) and \( x+y>1 \), the solution set isn't just one inequality's shaded region, but where both conditions are true. Hence, it's critical to accurately shade the correct regions for all individual inequalities first. Only the common shaded area represents the true solution set. It is a practical depiction of constraints in real-world problems such as budget limits, resource allocations, and design specifications where multiple conditions must be met simultaneously.