Problem 3 - Precalculus: Functions and Graphs

Question

Sketch the graph of the Inequality. $$2 x+3 y \geq 2 y+1$$

Step-by-Step Solution

Answer

Solid line \(2x + y = 1\) and shade above the line.

Step 1: Simplify the Inequality

Start by simplifying the inequality \(2x + 3y \geq 2y + 1\). To do this, subtract \(2y\) from both sides to isolate terms involving \(y\) on the left side. This results in \(2x + 3y - 2y \geq 1\), which simplifies to \(2x + y \geq 1\).

Step 2: Convert Inequality to Equation for Boundary Line

To sketch the graph of the inequality, first find the boundary line by converting the inequality to an equation. The boundary equation is \(2x + y = 1\). This line will help separate the plane into two regions.

Step 3: Determine Intercepts of the Boundary Line

To sketch the line, determine the intercepts. Set \(x = 0\) to find the y-intercept: \(2(0) + y = 1\) gives \(y = 1\). Set \(y = 0\) to find the x-intercept: \(2x + 0 = 1\) gives \(x = \frac{1}{2}\). Thus, the intercepts are (0, 1) and \(\left(\frac{1}{2}, 0\right)\).

Step 4: Sketch the Boundary Line

Plot the intercepts on a coordinate plane: (0, 1) and \(\left(\frac{1}{2}, 0\right)\). Draw a straight line through these points. The line represents the equation \(2x + y = 1\).

Step 5: Determine the Region for the Inequality

Choose a test point not on the line, such as (0,0), and substitute it into the inequality \(2x + y \geq 1\). This gives \(2(0) + 0 \geq 1\), which is false. This means the region that does not include the origin satisfies the inequality. Shade the region above the line.

Step 6: Finalize the Graph

Since the inequality is \(\geq\) (greater than or equal to), the boundary line is solid, indicating points on the line are also included in the solution. Thus, the final graph includes the solid line \(2x + y = 1\) and the shaded region above it.

Key Concepts

Linear Equations

A linear equation is an equation that makes a straight line when it is graphed on a coordinate plane. In its most basic form, the equation is written as \(ax + by = c\), where variables \(x\) and \(y\) are raised to the first power. In this exercise, the boundary line we derived from the inequality is represented by the linear equation \(2x + y = 1\). This line divides the coordinate plane into two distinct regions.

• Key components: coefficients for \(x\) and \(y\), constant term
• Represents a straight line when graphed

Understanding linear equations helps us graph these inequalities and determine boundaries for our solutions.

Coordinate Plane

The coordinate plane is a two-dimensional space formed by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis. These axes intersect at a point called the origin, denoted by (0,0). To graph an inequality, we need to understand how to navigate this plane. We used it here to plot the boundary line for our inequality. Each point on the plane is defined by a pair of numbers (x, y), indicating its horizontal and vertical positions.

• X-axis: horizontal line
• Y-axis: vertical line
• Origin: point (0,0)

The coordinate plane allows us to visualize solutions to equations and inequalities.

Boundary Line

In the context of graphing inequalities, a boundary line represents the critical line where an inequality switches from true to false (or vice versa). To understand in which region an inequality holds true, we convert the inequality to an equation to find the boundary line. For the given inequality, the boundary line is represented by \(2x + y = 1\). This line divides the plane into two regions.

• Solid line: \(\geq\) or \(\leq\), points on the line are included
• Dashed line: \(>\) or \(<\), points on the line are not included

By plotting this line and considering its position, we can determine where the inequality is satisfied.

Test Points

Using test points is a method to determine which region of the coordinate plane satisfies the inequality. After plotting the boundary line, we must identify which side of it is the solution to the inequality. A convenient test point is often (0,0), unless it lies on the boundary line itself. For the exercise, we used the test point (0,0) in the inequality \(2x + y \geq 1\). We found the inequality was not true for (0,0), meaning the solution region is on the opposite side of the boundary line from this point.

• Choose a point not on the line, like (0,0)
• Substitute into the inequality
• True: region includes the test point, False: region is opposite

Test points help definitively determine which region represents the solutions to the inequality.