Problem 30 - Algebra 2

Question

Graph each inequality. $$ y \geq \frac{1}{2} x-5 $$

Step-by-Step Solution

Answer

Graph the line \( y = \frac{1}{2}x - 5 \) and shade above the solid line.

Step 1: Write the Inequality in Slope-Intercept Form

The inequality is already given in slope-intercept form, which is \( y \geq \frac{1}{2}x - 5 \). Here, the slope \( m \) is \( \frac{1}{2} \), and the y-intercept \( b \) is \(-5\).

Step 2: Begin by Graphing the Related Equation

First, consider the equation \( y = \frac{1}{2}x - 5 \). Start by plotting the y-intercept at \( (0, -5) \) on the coordinate plane. This is where the line will cross the y-axis.

Step 3: Use the Slope to Find Another Point

From the point \( (0, -5) \), use the slope \( \frac{1}{2} \) to find another point on the line. The slope of \( \frac{1}{2} \) means "rise 1, run 2." So, from \( (0, -5) \), move up 1 unit and right 2 units to the point \( (2, -4) \).

Step 4: Draw the Boundary Line

Connect the points \((0, -5)\) and \((2, -4)\) with a straight line. Since the inequality is \( \geq \), draw a solid line to indicate that points on the line are included in the solution.

Step 5: Shade the Solution Region

Since we have \( y \geq \frac{1}{2}x - 5 \), shade the region above the line, because we are looking for all \( y \)-values greater than or equal to those on the line.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is useful for graphing and understanding inequalities. This form is expressed as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. It's particularly helpful because it directly shows how the line will behave on the coordinate plane, indicating where it starts and how steeply it rises or falls. In the inequality \( y \geq \frac{1}{2}x - 5 \), the slope \( m \) is \( \frac{1}{2} \). This tells us the line increases by 1 unit for every 2 horizontal units. The y-intercept \( b \) is \(-5\), showing the line crosses the y-axis at this point.To effectively graph this inequality, begin by plotting this y-intercept on the coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we plot points to represent solutions to equations and inequalities in mathematics. It consists of two axes: the horizontal \( x \)-axis and the vertical \( y \)-axis. Where these axes intersect is called the origin, denoted as \( (0,0) \). When graphing inequalities, like \( y \geq \frac{1}{2}x - 5 \), the coordinate plane allows us to visualize the relationship between two variables. By placing points according to their \( x \) and \( y \) values, we create a visual representation of the solution region. This helps in understanding whether a point satisfies the inequality as it falls within the shaded solution region. It's crucial to understand how to use this plane to accurately determine and represent numerical relationships.

Inequality Solutions

When dealing with inequalities, the solution includes a range of values rather than a single line of points like equations. In our example, \( y \geq \frac{1}{2}x - 5 \), the inequality suggests that every point where \( y \) is greater than or equal to \( \frac{1}{2}x - 5 \) is a solution. Points on the line itself are part of the solution because of the "greater than or equal to" (\( \geq \)) symbol. This makes the line solid, rather than dashed, confirming the inclusion of points on the boundary line as part of the solution. To visually identify solutions, we need to shade the correct part of the plane that meets the inequality's condition, taking care to cover the entire region above the boundary line.

Boundary Line

The boundary line is a key concept when graphing inequalities. It represents the line formed by the related equality equation, in this case, \( y = \frac{1}{2}x - 5 \). To graph this line:

• Start by plotting the y-intercept at \( (0, -5) \).
• Use the slope \( \frac{1}{2} \) to find another point, moving up 1 unit and over 2 units to \( (2, -4) \).

Once these points are plotted on the coordinate plane, connect them with a straight line. Since the inequality symbol is \( \geq \), draw this line solid to include the boundary line in the solution set. The line serves as a visual reference for determining which side of the line the solution region lies, guiding where to shade on the graph.

Shading Regions

Shading regions play a crucial role in graphing inequalities because they illustrate the solution set. In the context of \( y \geq \frac{1}{2}x - 5 \), shading shows all y-values above the boundary that satisfy the inequality.After drawing the solid boundary line, decide which side to shade:

• If the inequality symbol is "greater than" (\( \geq \)), shade above the line.
• For "less than" (\( \leq \)), shade below.

Testing a point can confirm your shading is correct. Choosing a simple point such as the origin \( (0, 0) \) can help: if it satisfies \( 0 \geq \frac{1}{2}(0) - 5 \), which it does not, then shade the opposite side. This technique visually communicates which values of \( y \) fulfill the inequality given any \( x \). Shading can be a powerful tool to quickly determine solutions.