Problem 72 - A Graphical Approach to College Algebra

Question

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned} &y \geq|x+2|\\\ &y \leq 6 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is the region above \( y = |x+2| \) and below \( y = 6 \).

Step 1: Understand the Inequalities

We have two inequalities: 1) \( y \geq |x+2| \) and 2) \( y \leq 6 \). For the first inequality, \( y \geq |x+2| \), it means the area above or on the graph of \( y = |x+2| \) should be shaded. For the second inequality, \( y \leq 6 \), the area below or on the horizontal line \( y = 6 \) should be shaded.

Step 2: Graph the Absolute Value Function

Graph \( y = |x+2| \) on your calculator. This forms a V-shape with a vertex at \((-2,0)\). The graph opens upwards and increases without bound as \(x\) moves away from \(-2\).

Step 3: Apply the First Inequality

Shade the region above or on the graph of \( y = |x+2| \). This represents the solution to the inequality \( y \geq |x+2| \).

Step 4: Graph the Horizontal Line

Graph the line \( y = 6 \). This is a horizontal line that intersects the y-axis at 6.

Step 5: Apply the Second Inequality

Shade the region below or on the line \( y = 6 \). This represents the solution to the inequality \( y \leq 6 \).

Step 6: Find the Overlapping Region

Identify the region that is shaded for both inequalities. This is where the solution to the system \( y \geq |x+2| \) and \( y \leq 6 \) lies. This region forms a band above the absolute value function and below the line \( y = 6 \).

Key Concepts

Absolute Value Function

The absolute value function is an important concept in mathematics, often represented as \( y = |x| \). This type of function creates a 'V' shape graph, characterized by its vertex and symmetry. In this exercise, we are working with the function \( y = |x+2| \). This function has its vertex at the point \((-2,0)\). This means that the graph has been shifted two units to the left from the standard absolute value function \( y = |x| \). When graphing \( y = |x+2| \), it is crucial to understand that the V-shape tells us that as \( x \) moves away from \(-2\), the value of \( y \) increases.

• Vertex: The point where the graph changes direction \((-2, 0)\).
• Shape: V-shaped and symmetrical around its vertex.
• Direction: Opens upwards.

Understanding this will help you graph inequalities involving absolute values more accurately.

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing and solving mathematical problems like inequalities. It allows you to accurately graph functions and determine intercepted points without performing manual calculations.When using a graphing calculator to solve the problem of graphing \( y \geq |x+2| \) and \( y \leq 6 \), here is what you need to do:

• Input the absolute value function \( y = |x+2| \) into the calculator. Look for the 'absolute value' functionality, often found in the math menu.
• Graph the horizontal line \( y = 6 \). Most graphing calculators have a straightforward way to input straight lines.
• Use the shading capabilities of the graphing calculator to highlight the solution region for each inequality.

This method not only makes solving such problems more accessible but also helps you visualize the relationships between different inequalities.

Shading Regions

Shading regions on a graph is an effective way to represent solutions to inequalities. It visually distinguishes areas that satisfy the inequality conditions. For the inequality \( y \geq |x+2| \), the shaded region includes all the points on or above the V-shaped graph of the absolute value function.

• Above the function: Indicates the region where \( y \) is greater than or equal to \( |x+2| \).
• Below the function: Would indicate \( y \leq |x+2| \), which is not required here.

For \( y \leq 6 \), you shade below or on the line \( y = 6 \), which limits the range for \( y \).

• Below the line: Indicates \( y \) values up to and including 6.

Together, the overlapping area of both shaded regions represents the solution to the system of inequalities.

Systems of Inequalities

Systems of inequalities involve solving more than one inequality at a time and finding a common solution area that satisfies all conditions. In this case, we are looking at \( y \geq |x+2| \) and \( y \leq 6 \). To solve this system, the graph must highlight the simultaneous shading of the regions:

• Graph the V-shaped absolute value function \( y = |x+2| \) and shade above it.
• Graph the horizontal line \( y = 6 \) and shade below it.

The region where these shadings overlap designates the set of solutions for the system. This region forms a compact band, illustrating all possible solutions that satisfy both inequalities.By recognizing the intersection of shaded areas, one can accurately interpret the solution to such a system, making the concept a potent tool in mathematical graphing.