Problem 31 - Algebra A Combined Function

Question

Graph each inequality. $$ y \geq \frac{1}{2} x-4 $$

Step-by-Step Solution

Answer

Plot the line \( y = \frac{1}{2}x - 4 \) and shade above it.

Step 1: Identify the Inequality Type

The inequality given is \( y \geq \frac{1}{2}x - 4 \). This inequality involves a linear function and "greater than or equal to", which means that the graph will include a solid line for the boundary of the region and the region above the line.

Step 2: Convert to Equation for Boundary Line

To graph the inequality, first graph the line that forms the boundary by turning the inequality into an equation. The equation of the boundary line is \( y = \frac{1}{2}x - 4 \).

Step 3: Plotting Points to Draw the Line

Calculate two or more points on the line. For example, when \( x = 0 \), \( y = \frac{1}{2}(0) - 4 = -4 \). The point is \( (0, -4) \). When \( x = 2 \), \( y = \frac{1}{2}(2) - 4 = -3 \). The point is \( (2, -3) \). Plot these points on a graph.

Step 4: Draw the Boundary Line

Draw a solid line through the points plotted. Since the inequality symbol \( \geq \) includes equality, the boundary line is solid. This line represents all points \( (x, y) \) where \( y = \frac{1}{2}x - 4 \).

Step 5: Shade the Correct Region

Since the inequality is \( y \geq \frac{1}{2}x - 4 \), we shade the region above the line. This represents all the points for which \( y \) is greater than or equal to the line \( y = \frac{1}{2}x - 4 \).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe a straight line when graphed on a coordinate plane. They take the form \( y = mx + b \), where "\( m \)" is the slope and "\( b \)" is the y-intercept. In our case, the boundary line equation is \( y = \frac{1}{2}x - 4 \), representing a line with a slope of \( \frac{1}{2} \) and a y-intercept of -4.
To plot a linear equation, follow these steps:

• Identify the slope and y-intercept from the equation.
• Start at the y-intercept point on the graph.
• Use the slope to find another point on the line. For each unit you go right (positive x-direction), move up by the value of the slope (if it's positive) or down (if it's negative).

By connecting these identified points, you create the straight line that represents the linear equation.

Shading Regions

Shading represents areas on a graph where the solutions to an inequality exist. When you graph an inequality, the line divides the plane into two halves, and only one of them contains solutions to the inequality.

For the inequality \( y \geq \frac{1}{2}x - 4 \), the shaded region is above the boundary line. You determine which side to shade by testing a point not on the boundary. If it satisfies the inequality, that is the side to shade.

• Choose a test point, commonly \( (0,0) \), unless it’s on the line.
• Substitute the test point into the inequality.
• If the inequality holds true with the test point, shade the side containing that point.
• If not, shade the opposite side.

This process ensures you cover all correct solutions that satisfy the inequality.

Boundary Line

The boundary line in the graph of an inequality helps you understand the limits of the area you must consider. It demarcates the set of points where the equation part of the inequality holds true, \( y = \frac{1}{2}x - 4 \) in our problem.

The boundary line will be solid if the inequality symbol is \( \geq \) or \( \leq \), as these symbols include the equality condition. This means any point on the line is a solution to the inequality as well.

• In the graph, a solid line means the points on the line are included in the solution set.
• Whereas, a dashed line indicates that points on the line are not included because the inequality is only "less than" (<) or "greater than" (>).

This distinction is crucial for accurately representing inequalities graphically.

Inequality Symbols

Inequality symbols are critical in understanding the nature and solutions of inequalities. The symbols "<", "<=", ">", and ">=" indicate whether solutions include the boundary and which side of the boundary is correct.

In our example \( y \geq \frac{1}{2}x - 4 \):

• The symbol \( \geq \) signifies "greater than or equal to".
• This inequality includes the boundary line, hence the line is solid.
• Solutions are represented by points on and above the line, covering both equal and greater values of \( y \).

Understanding these symbols ensures you correctly interpret and graph the solutions, capturing all possible answers to the inequality.