Problem 38 - Algebra 2

Question

You can graph inequalities by using the SHADE (command located in the DRAW menu. Enter two functions. \(\bullet\) The first function defines the lower boundary of the shaded region. If the inequality is " \(y \leq,\) use the Ymin window value as the lower boundary. \(\bullet\) The second function defines the upper boundary of the region. If the inequality is " \(y \geq,\) "use the Ymax window value as the upper boundary. Graph each inequality. $$ x-7 \leq y $$

Step-by-Step Solution

Answer

Graph and shade the region above the line \( y = x - 7 \).

Step 1: Identify the Inequality

The given inequality is \( x - 7 \leq y \). This indicates that the region to be shaded is where \( y \) is greater than or equal to \( x - 7 \). Thus, \( x - 7 \) is the lower boundary of the shaded area when graphed.

Step 2: Determine Lower Boundary Function

From the inequality \( x - 7 \leq y \), we identify the lower boundary function as \( y = x - 7 \). This will be the first function to graph and will serve as the linear boundary below which shading will occur.

Step 3: Determine Upper Boundary Function

Since the inequality is expressed as \( x - 7 \leq y \), which translates to \( y \geq x - 7 \), and there is no explicit upper limit, we use the maximum value of the Y window function as the upper boundary for the shaded region. This value is usually set in the graphing calculator's window settings.

Step 4: Set Up the Graphing Window

Adjust the graphing window on your calculator. For this exercise, you set the Ymin to the minimum value of \( y \) that you want the graph to display and the Ymax to an upper value based on the calculator settings or problem constraints. The Xmin and Xmax settings will determine the horizontal viewing range of the graph.

Step 5: Input Functions into Calculator

Enter the lower boundary as the first function: \( y_1 = x - 7 \). Depending on the specific calculator instructions for shading or drawing inequalities, ensure this function is set to account for the region where \( y \geq x - 7 \). Use the Ymax value in settings to create the upper boundary.

Step 6: Graph and Shade the Region

After entering the functions into the graphing calculator, access the draw menu and use the SHADE command. Shade the region that falls between \( y = x - 7 \) and the Ymax line, which is the top of the viewing window. The area between these two lines represents the solution to the inequality \( x - 7 \leq y \).

Key Concepts

Linear Inequalities

Linear inequalities are like linear equations, but instead of an equal sign, they use inequality symbols such as \(<, >, \leq,\) or \(\geq\). These symbols show relationships between variables and help determine the range of possible solutions.
For the inequality \(x - 7 \leq y\), it indicates that the area above the line \(y = x - 7\) on a graph is included in the solution. This line becomes a "boundary," and because of the symbol \(\leq\), the solutions lie on or above it.
Understanding linear inequalities involves interpreting these symbols and knowing how they affect the graph's region. Recognizing this is crucial for solving and graphing.

• \(x - 7 \leq y\) can be rewritten with \(y\) as the subject: \(y \geq x - 7\).
• The solutions are all the points \((x, y)\) that lie above or on this line when graphed on a coordinate plane.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing functions and solving inequalities. It helps make abstract concepts more concrete by displaying them visually.
To graph inequalities, the calculator uses commands like SHADE, allowing you to fill areas between boundaries on the graph.
Before graphing, input any functions you'll need. For this example, you input \(y_1 = x - 7\) as the first function. Setting graphing window parameters correctly ensures you clearly see the required portions of the graph.

• Settings like Xmin and Xmax define the horizontal span of visibility on the calculator screen.
• Ymin and Ymax set vertical boundaries, crucial when graphing inequalities with a specific upper or lower boundary.

Using these features correctly will allow you to successfully depict and solve inequality problems.

Shaded Regions

When working with inequalities, shaded regions on a graph illustrate where solutions lie.
The solution area indicates all possible \((x, y)\) values satisfying the inequality. For \(x - 7 \leq y\), it’s the area above the line \(y = x - 7\).
Shading is an effective way to show these areas visually, providing a comprehensive understanding of inequalities beyond just the line.
To show shaded regions:

• Identify the boundary line, which here is \(y = x - 7\).
• Determine if the shading is above or below the line, depending on the inequality symbol (\leq or \geq for above).
• Use these facts to cover the appropriate region on your graphing calculator.

Effective shading helps easily spot all solutions meeting the criteria set by the inequality.

Graphing Functions

Graphing functions is about plotting a set of possible solutions that satisfy a function equation.
To graph the function \(y = x - 7\), find various points \((x, y)\) that satisfy this equation. Plot these points and draw the line through them.
Functional graphing requires understanding function behavior:

• Linear functions like \(y = x - 7\) graph as straight lines.
• The gradient (slope) here is 1, indicating a diagonal line rising one unit up for every unit it moves right.

By employing graphing functions, you can visualize how lines interact in conjunction with inequalities, helping you see how the solution space forms around them.