Problem 47 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x^{2}+y^{2} \leq 16 \\ x+y>2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set is the region inside and on the circle \(x^2 + y^2 = 16\) and above the line \(x + y = 2\).

Step 1: Graph the Circle

Start by graphing the circle \(x^2+y^2\leq16\). This is a circle centered at the origin (0,0) with a radius of 4. The inequality symbol \(\leq\) indicates that the solution includes the points on the circle as well as those inside it. Draw this on your graph.

Step 2: Graph the Line

Next, graph the line \(x+y>2\). This is a straight line which intercepts the y axis at (0,2) and the x axis at (2,0). The inequality symbol \(>\) indicates that the solution includes the points above the line but not on the line itself. Draw this on your graph, making sure to indicate that the area above the line is the solution set.

Step 3: Find the Solution Set

The solution set of the system of inequalities is the region where both conditions are met. This is the area within the circle (including the perimeter) which also lies above the line. Shade this area to highlight it.

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where the variable is raised to the second power, such as in the equation \(x^2 + y^2 \leq 16\). Essentially, you're dealing with a parabola or, in this case, a circle, because the variable \(x\) and \(y\) are squared and added. Graphing a quadratic inequality begins with sketching the related 'equal to' shape, which here is the circle with radius 4 centered at the origin.

The inequality sign \(\leq\) tells us that the solutions include not just points on the circle but also all the points inside it. To show this on a graph, we typically shade the area that contains the solutions. It’s vital to recognize the difference between a 'less than' and a 'less than or equal to' inequality, where the latter includes the boundary (the circle itself) as part of the solution set.

Linear Inequalities

Linear inequalities are algebraic expressions that involve a first-degree polynomial, which when graphed, forms a straight line. The example given, \(x + y > 2\), represents such an expression. To graph this inequality, one would first graph the corresponding equation \(x + y = 2\). This can be done by locating the intercepts on the axes: (0,2) on the y-axis and (2,0) on the x-axis.

Afterwards, since the inequality is strict (\(>\)), the area above the line is shaded, indicating that it contains the solutions. It's important to note that the line itself is not part of the solution set, so it is typically represented with a dashed line to show that the points on the line do not satisfy the inequality.

Inequality Graphing

Inequality graphing is a visual way to represent the solution sets of inequalities on a coordinate plane. The process generally involves two main steps: graphing the boundary, which is the equality part of the inequality, and then determining and shading the area that represents the solution set.

With systems of inequalities, multiple shaded regions must be considered, and the shared region between them is the solution to the system. It's essential to differentiate solid lines, which include the boundary points as solutions, and dashed lines, which exclude the boundary points. Use of shading or patterns helps indicate where the inequalities overlap and is vital for correctly identifying the solution set.

Solving Inequalities

Solving inequalities means finding all the values of the variable that make the inequality true. Unlike equalities (equations), solutions to inequalities are frequently entire ranges of numbers or areas in a graph.

The process mainly involves isolating the variable after dealing with any simplification that needs to be done. Remember that reversing the inequality sign is necessary when you multiply or divide both sides by a negative number. In the case of systems of inequalities, the solution consists of the values that satisfy all inequalities in the system, which, when graphed, is the intersection of their individual solution sets.