Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Graph each inequality. $$2 x>3-4 y$$

Step-by-Step Solution

Answer

The inequality graph is a dashed line for \(y = \frac{1}{2}x - \frac{3}{4}\), shaded above.

Step 1: Rewrite the Inequality

First, we need to rearrange the given inequality to solve for \(y\). Start by bringing all terms involving \(y\) to one side of the inequality:\[2x - 3 > -4y\] Now, add \(4y\) to both sides to isolate the \(y\)-terms on one side of the inequality:\[2x - 3 + 4y > 0\]

Step 2: Solve for y

Next, we need \(y\) by itself on one side of the inequality. Rearrange the expression from Step 1 by subtracting \(2x - 3\) from both sides:\[4y > 2x - 3\]Now, divide every term by 4 to solve for \(y\):\[y > \frac{1}{2}x - \frac{3}{4}\]

Step 3: Graph the Boundary Line

To graph the inequality \(y > \frac{1}{2}x - \frac{3}{4}\), first graph the line \(y = \frac{1}{2}x - \frac{3}{4}\) as a boundary. This line has a slope of \(\frac{1}{2}\) and a y-intercept of \(-\frac{3}{4}\). Since the inequality is strict (\(>\), not \(\geq\)), this line should be dashed to indicate it is not part of the solution set.

Step 4: Determine the Shaded Region

The inequality \(y > \frac{1}{2}x - \frac{3}{4}\) tells us to shade above the line because \(y\) is greater than the expressions represented by the line. You can test a point not on the line, such as (0,0), to verify if it satisfies the inequality (here, it does). The entire region above the dashed line is the solution area.

Key Concepts

Graphing Inequalities

Graphing inequalities allows us to visually represent all solutions to an inequality within a coordinate plane. This process usually involves a few steps.

• First, rewrite the inequality, as seen in the problem, to have the variable of interest (often y) isolated on one side.
• Next, plot the corresponding boundary line of the inequality, initially ignoring the inequality symbol. For example, graph the equality \( y = \frac{1}{2}x - \frac{3}{4} \).
• If the inequality symbol is either \( \leq \) or \( \geq \), draw a solid line because the line is part of the solution. If it's \( < \) or \( > \), like in our case here, use a dashed line to indicate the line itself is not included in the solution.
• Finally, choose a test point to determine which side of the line to shade, indicating the solution set. This helps in understanding which region satisfies the inequality.

Understanding graphing inequalities is key to visualizing and solving inequalities in contexts like optimization problems and systems of inequalities.

Linear Equations

A linear equation forms the backbone of graphing inequalities, exemplifying a straight line on the coordinate plane. It appears in the slope-intercept form as \( y = mx + b \), where:

• \( m \) is the slope representing the steepness and direction of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

For our problem, the derived boundary line is \( y = \frac{1}{2}x - \frac{3}{4} \). Here:

• The slope \( \frac{1}{2} \) signifies the line rises one unit for every two units it moves along the x-axis.
• The y-intercept of \( -\frac{3}{4} \) indicates the point \( (0, -\frac{3}{4}) \) on the graph.

Understanding linear equations is crucial since they form the boundary for many inequality graphs and serve as a stepping stone to more complex algebraic concepts.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph equations and inequalities. It consists of two perpendicular axes:

• The x-axis is horizontal, while the y-axis is vertical.

Every point on this plane is identified by a pair of numbers \((x, y)\), which describe its horizontal and vertical distances from the origin \((0,0)\).
In graphing an inequality like \( y > \frac{1}{2}x - \frac{3}{4} \), the coordinate plane provides the space to visualize both the boundary line and the solutions. Once the line is drawn, the plane is effectively divided into two regions, helping us see where the inequality holds.
By shading the correct region - above the dashed line here, we emphasize the solution area in relation to the initial inequality.

Solution Region

The solution region is the area on the graph that contains all the solutions to the inequality. After plotting the boundary line for our inequality \( y > \frac{1}{2}x - \frac{3}{4} \), we determine which side of the line contains the correct solutions.
To find the solution region, use a test point not on the line, such as \( (0, 0) \). Substitute this point into the inequality. If the inequality holds true, then this side is part of your solution region. Otherwise, the solution is on the opposite side.
In this case, substituting \( (0, 0) \) results in an inequality that is true, confirming that the region above the line is where the inequality holds. Moreover, utilizing shading helps visually express this region, capturing the essence of where the inequality solutions are gathered. So, shading is not only a graphical annotation but also a conceptual tool to understand inequalities better.