Problem 35 - Algebra A Combined Function

Question

Explain how to decide which region to shade to show the solution region of the following system. $$ \left\\{\begin{array}{l} x \geq 3 \\ y \geq-2 \end{array}\right. $$

Step-by-Step Solution

Answer

Shade the region where \( x \geq 3 \) and \( y \geq -2 \) overlap.

Step 1: Graph the Inequality x ≥ 3

To graph the inequality \( x \geq 3 \), draw a vertical line at \( x = 3 \). Since the inequality is \( \geq \), shade the region to the right of the line because these values satisfy the condition \( x \geq 3 \).

Step 2: Graph the Inequality y ≥ -2

Next, graph the inequality \( y \geq -2 \) by drawing a horizontal line at \( y = -2 \). Since the inequality is \( \geq \), shade the region above the line, as these values are \( y \geq -2 \).

Step 3: Find the Solution Region

The solution region is where the shaded areas from both inequalities overlap. This area represents all points that satisfy both \( x \geq 3 \) and \( y \geq -2 \). Shade this overlapping region to show the solution to the system of inequalities.

Key Concepts

Graphing Inequalities

To solve a system of inequalities, you start by graphing each inequality on the same coordinate plane. Understanding how to graph inequalities is essential as it sets the foundation for finding the solution region. Each inequality has its own set of rules for graphing. For the inequality \( x \geq 3 \), you'll draw a vertical line at \( x = 3 \). The vertical line represents all points where \( x \) equals 3. Then, because we have a "greater than or equal to" inequality (\( \geq \)), you will shade in the area to the right of this line. This shaded area includes all points with \( x \) values that are greater than or equal to 3.Similarly, for the inequality \( y \geq -2 \), a horizontal line is drawn at \( y = -2 \). This line shows all the points where \( y \) equals -2. You will then shade above this line because \( y \geq -2 \) includes all points where \( y \) is greater than or equal to -2. The key to graphing inequalities effectively is drawing the lines accurately and shading the correct regions based on the inequality signs.

Solution Region

The concept of a solution region is central when dealing with systems of inequalities. Once you have graphically represented each inequality, your task is to identify the solution region.The solution region is where the shaded areas of all inequalities intersect or overlap. This overlapping area represents all points that satisfy every inequality in the system. It's vital to accurately shade the solution region, as it showcases the combination of all conditions being met at the same time.In our exercise, after graphing both \( x \geq 3 \) and \( y \geq -2 \), the solution region is found to the right of the line \( x = 3 \) and above the line \( y = -2 \). Any point in this region will fulfill both conditions simultaneously, making it an optimal visual representation of solutions to the system.Ensuring the solution region is accurately portrayed reinforces the understanding of not just one, but a collection of inequalities working together in unison.

Shading Technique

Performing the correct shading technique is a critical skill when working with inequalities on a graph. Shading helps to visually emphasize which areas of the graph satisfy the inequality.The direction in which you shade depends on the inequality symbol. For instance:

• If the inequality is \( \geq \) or \( \leq \), you need a solid line because it includes the border (the line) itself.
• If it is \( > \) or \( < \), then a dashed line is used since it does not include the points on the line.

In our example, since both inequalities involve the "greater than or equal to" sign, both lines are drawn solid, and:

• The region to the right of \( x = 3 \) is shaded to show that any \( x \) value greater than or equal to 3 is valid.
• The region above \( y = -2 \) is shaded to indicate all \( y \) values greater than or equal to -2 are part of the solution.

The intersection of these shaded regions forms the complete solution set. Using the right shading techniques ensures clarity and precision in graphically representing systems of inequalities.