Problem 52 - Elementary Algebra

Question

$$ \text { For Problems 37-56, graph each linear inequality. (Objective } 3 \text { ) } $$ $$ y \geq-\frac{1}{2} x+1 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Inequality Format

The inequality given is in the format \( y \geq mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The inequality is \( y \geq -\frac{1}{2}x + 1 \). Here, \( m = -\frac{1}{2} \) and \( b = 1 \).

Step 2: Identify the Boundary Line

The boundary line of the inequality is given by the equation \( y = -\frac{1}{2}x + 1 \). This line will help us determine the region to shade for the inequality.

Step 3: Plot the Boundary Line

Start by plotting the y-intercept (0,1) on the graph. Then, use the slope \( m = -\frac{1}{2} \) to find another point. From (0,1), move down 1 unit and right 2 units to plot the next point (2,0). Connect these points with a solid line, since the inequality is \( \geq \), which includes the boundary.

Step 4: Determine the Shaded Region

Since the inequality is \( y \geq -\frac{1}{2}x + 1 \), we will shade the region above the line. To verify, pick a test point not on the line, like (0,2). Substitute this into the inequality: \( 2 \geq -\frac{1}{2}(0) + 1 \) which simplifies to \( 2 \geq 1 \), a true statement, confirming that the region above the line is correct.

Key Concepts

Linear Equations

Linear equations are vital in understanding math as they represent relationships between two variables with a constant rate of change. A linear equation is typically written in the form \( y = mx + b \). In this expression:

• \( y \) and \( x \) are variables.
• \( m \) is the slope, showing the rate at which \( y \) changes with respect to \( x \).
• \( b \) is the y-intercept, which indicates the point where the line crosses the y-axis.

Linear equations create graph lines that are straight and can extend infinitely in both directions. Graphing these lines on the coordinate plane aids in visualizing how one variable changes in relation to the other.

Slope-Intercept Form

The slope-intercept form, \( y = mx + b \), is an efficient way to graph linear equations. It makes identifying a line's slope and y-intercept straightforward. Here's how each element plays a role:

• **Slope (\( m \)):** This tells you how steep a line is. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
• **Y-Intercept (\( b \)):** This is where the line touches the y-axis. It's the starting point when you begin graphing a line using the slope.

For example, in the equation \( y \geq -\frac{1}{2}x + 1 \), the slope \( m = -\frac{1}{2} \) indicates a decreasing line, and the y-intercept is at (0,1). You could start plotting from this point and use the slope to find more points, helping you line up the plot on a graph.

Shaded Region

When graphing linear inequalities like \( y \geq -\frac{1}{2}x + 1 \), it's crucial to identify which side of the line to shade. This shading indicates all the possible solutions for the inequality. The steps to determine the shaded region include:

• **Graph the Boundary Line:** First, graph the line \( y = -\frac{1}{2}x + 1 \) using the slope and intercept.
• **Test a Point:** Choose a point not on the line, such as (0,2). Substitute this into the inequality to see if it holds true.
• **Shade the Correct Side:** If the test point satisfies the inequality, shade that side of the line. With \( y \geq -\frac{1}{2}x + 1 \), the region above the line meets the condition, so it is shaded.

Shading helps visualize the solution set and shows all the combinations of \( x \) and \( y \) that work in the inequality.

Boundary Line

The boundary line in graphing linear inequalities is derived from converting the inequality to an equation by using equality instead of inequality, like in \( y = -\frac{1}{2}x + 1 \). This line:

• Serves as a divider between the solution region and non-solution region on the graph.
• Is typically solid when the inequality is "greater than or equal to" (\( \geq \)) or "less than or equal to" (\( \leq \)). A solid line means points on the line satisfy the inequality.
• Is dotted when the inequality is "greater than" (\( > \)) or "less than" (\( < \)), indicating points on the line don't satisfy the inequality.

By plotting the boundary line and shade the region on the correct side, you will effectively represent the inequality's solution set on the graph.