Problem 37 - Thinking Mathematically

Question

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x+2 y \leq 4 \\ y \geq x-3\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to this system of inequalities is the overlapping shaded area on the graph that is under the line of the first equation and above the line of the second equation.

Step 1: Graph the first inequality

The first inequality is \(x + 2y \leq 4\). This can be written as \(y \leq 4/2 - x/2\), or \(y \leq 2 - 0.5x\). Draw this line on the graph, marking it dashed because points on the line are not included in the solution (due to '<' sign). Shade the area below the line since the inequality is less than or equal to.

Step 2: Graph the second inequality

The second inequality is \(x - y \leq 3\) or equivalently \(y \geq x - 3\). Draw another line on the graph, mark it solid because points on the line are included in the solution (due to '>=' sign). Shade the area above the line since the inequality is greater than or equal to.

Step 3: Identify the solution area

The solution to the system of inequalities is the area where both shaded parts overlap. This is the region that satisfies both inequalities simultaneously.

Key Concepts

Inequality Graphing

Understanding how to graph inequalities is a fundamental skill in algebra that can help visualize the solutions to a given problem. Unlike equations, inequalities do not just have one solution, but rather a range of possible solutions. When graphing an inequality that involves two variables, like the one in our exercise \(x + 2y \leq 4\), the solution is represented as a region on the coordinate plane rather than a single line.

To graph the inequality, we first convert it into an equation by ignoring the inequality sign and then plot the resulting line. In the provided exercise, \(y \leq 2 - 0.5x\) converts to the line \(y = 2 - 0.5x\). Depending on the inequality sign, we use a dashed line to indicate that points on the line are not part of the solution, or a solid line to show that they are included. For \(y \leq\) or \(y <\), we shade the area below the line, and for \(y \geq\) or \(y >\), we shade the area above the line, representing all the points that satisfy the inequality.

Linear Inequalities

Linear inequalities, such as \(x + 2y \leq 4\) and \(y \geq x - 3\) in our exercise, are simply inequalities that can be expressed with linear expressions. These inequalities can have one or more variables and generally form a straight line when the equality holds.

The graphing process involves similar steps to graphing linear equations, with the key difference being the need to indicate whether points on the line are part of the solution set. The notations '<' and '>' inform us that the line itself is not part of the solution, represented by a dashed line, while '\(\leq\)' and '\(\geq\)' include the line in the solution, shown with a solid line. The final step is to shade the appropriate side of the line: for instance, if the variable is greater than the expression, we shade above the line, and if it's lesser, we shade below.

System of Inequalities Solution

A system of inequalities, like the set we're looking at in our exercise, will often consist of multiple inequalities that need to be satisfied simultaneously. The solution to such a system is the intersection of the solutions to the individual inequalities, meaning the area where the shaded regions of both inequalities overlap on the graph.

To find this solution set, we graph each inequality separately on the same coordinate plane and look for the common shaded area. For the system given as \(\left\{\begin{array}{l}x+2y \leq 4 \ y \geq x-3\end{array}\right.\), the solution area is the portion of the plane that is shaded below the line for \(x + 2y \leq 4\) and above the line for \(y \geq x - 3\). It is this overlapping region that forms the set of ordered pairs satisfying both inequalities.

Algebraic Inequalities

Algebraic inequalities are mathematical expressions that show the relationship between two values, using inequality signs like '<', '>', '\(\leq\)', and '\(\geq\)'. They are used to express that one algebraic expression is less than, greater than, less than or equal to, or greater than or equal to another expression.

These inequalities are paramount in situations where a range of potential solutions is possible, not just a single answer. They describe conditions and constraints and can be found in various real-world scenarios, from business models to scientific calculations. When solving algebraic inequalities, we follow similar steps as with equations by isolating one variable on one side of the inequality while maintaining the balance of the inequality by performing the same operations on both sides. However, we need to remember that if we multiply or divide by a negative number, we must reverse the direction of the inequality sign.