Problem 40 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Graph and shade below the line from (3,0) to (0,1).

Step 1: Identify the Inequality Form

The inequality is given as \( x + 3y < 3 \). This is a linear inequality in two variables (\(x\) and \(y\)). The inequality sign is "<", indicating that we need to find the region where \( x + 3y \) is less than 3.

Step 2: Convert to Equation for Boundary Line

To graph the inequality, we first need to find the boundary line by converting the inequality into an equation: \( x + 3y = 3 \). This line will help us determine the region of the plane to shade.

Step 3: Find Intercepts of the Line

Find the x- and y-intercepts of the boundary line \( x + 3y = 3 \). To find the x-intercept, set \( y = 0 \) and solve for \( x \): \( x = 3 \). To find the y-intercept, set \( x = 0 \) and solve for \( y \): \( 3y = 3 \Rightarrow y = 1 \). These intercepts are points (3,0) and (0,1).

Step 4: Graph the Boundary Line

Plot the points (3,0) and (0,1) on a coordinate plane and draw a dashed line through these points. A dashed line indicates that the points on the line are not included in the solution (since the inequality is "<" and not "≤").

Step 5: Shade the Solution Region

Choose a test point not on the line, such as the origin (0,0), and substitute it into the inequality \( x + 3y < 3 \). This gives \( 0 + 3(0) < 3 \Rightarrow 0 < 3 \), which is true, so shade the region that includes the origin. This region represents all points \((x, y)\) that satisfy \( x + 3y < 3 \).

Key Concepts

Graphing Inequalities

When it comes to graphing inequalities, it's a straightforward process if you break it down into simple steps. Inequalities like \( x + 3y < 3 \) define a region in the coordinate plane where the inequality holds true. The main goal is to identify and shade this region.

• First, convert the inequality into an equation by replacing the inequality symbol with an equals sign. This helps in finding the boundary line.
• Next, determine whether to draw a solid or a dashed line. Use a dashed line for '<' or '>', and a solid line for '≤' or '≥' to show whether points on the line are included in the solution.
• Finally, choose a test point to decide which side of the line to shade. A common choice is the origin \((0, 0)\), if it’s not on the boundary line.

Follow these steps, and you'll be able to graph linear inequalities with confidence!

Coordinate Plane

The coordinate plane is a two-dimensional space formed by two perpendicular lines called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. Together, they divide the plane into four quadrants.

• The point where the axes intersect is the origin \((0, 0)\).
• Every point in the plane can be described with an ordered pair \((x, y)\), where \(x\) and \(y\) represent distances along the x-axis and y-axis, respectively.
• Understanding the coordinate plane is crucial for graphing since it provides the framework where all plotting takes place.

Remember, practice will help you become more familiar with navigating and using the coordinate plane effectively.

Boundary Line

A boundary line plays an essential role in graphing linear inequalities. Once you have converted the inequality to an equation, this line helps visualize the split between regions.

• To draw the boundary line, find points that satisfy the equation. A good approach is to determine the x- and y-intercepts.
• For example, with the equation \(x + 3y = 3\), you find intercepts where the line crosses the axes: (3,0) on the x-axis and (0,1) on the y-axis.
• After plotting these points, draw the line. Remember: use a dashed line for '<' or '>', indicating that points on the line are not part of the solution.

Mastering the concept of the boundary line gives you a strong foundation for solving linear inequalities.

Solution Region

The solution region is the part of the coordinate plane where all points satisfy the linear inequality.

• To identify the solution region, pick a test point not on the boundary line, like \((0, 0)\), and plug it into the original inequality.
• If the inequality holds true with this test point, then the region containing this point is the solution region.
• Shade this area to visually indicate where the inequality \(x + 3y < 3\) is true.

By shading the solution region, you not only show the solution graphically but also enhance your understanding of the inequality's implications across the plane.