Problem 10 - A Graphical Approach to College Algebra

Question

Graph each inequality. Do not use a calculator. $$2 x>3-4 y$$

Step-by-Step Solution

Answer

Rearrange as \(y > -\frac{1}{2}x + \frac{3}{4}\), graph the dashed line, shade above.

Step 1: Rearrange the Inequality

The given inequality is \(2x > 3 - 4y\). To make it easier to graph, we first rearrange it into the form \(y = mx + b\). Start by adding \(4y\) to both sides: \(2x + 4y > 3\).

Step 2: Solve for y

Continue rearranging the inequality to solve for \(y\). Subtract \(2x\) from both sides to get \(4y > -2x + 3\). Next, divide all terms by \(4\): \(y > -\frac{1}{2}x + \frac{3}{4}\).

Step 3: Identify Slope and Y-Intercept

The inequality \(y > -\frac{1}{2}x + \frac{3}{4}\) shows a slope \(m = -\frac{1}{2}\) and y-intercept \(b = \frac{3}{4}\). These will guide us in sketching the boundary line.

Step 4: Draw the Boundary Line

The boundary line, \(y = -\frac{1}{2}x + \frac{3}{4}\), is dashed since the inequality does not include equality (\(>\) instead of \(\geq\)). Start by plotting the y-intercept \((0, \frac{3}{4})\) and use the slope \(-\frac{1}{2}\) to find another point: from \(x = 0\), move right 2 units and down 1 unit to plot \((2, \frac{1}{4})\). Draw a dashed line through these points.

Step 5: Shade the Solution Region

Choose a test point not on the line, like \((0,0)\). Substitute into the inequality \(0 > -\frac{1}{2}(0) + \frac{3}{4}\). Since \(0 > \frac{3}{4}\) is false, shade the opposite side of the line from \((0,0)\), representing the region where \(y > -\frac{1}{2}x + \frac{3}{4}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a powerful tool in graphing inequalities. It's represented as \(y = mx + b\). Here, \(m\) stands for the slope, and \(b\) is the y-intercept.

• The slope \(m\) indicates the steepness or incline of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
• The y-intercept \(b\) is the point where the line crosses the y-axis, allowing us to easily start plotting our graph.

When dealing with inequalities, this form helps in identifying the boundary line clearly. The slope-intercept form not only aids in plotting a line but also in understanding the direction in which the inequality is true. Translating the given inequality into this format simplifies the graphing process and gives a clear picture of how to proceed.

Boundary Line

In graphing an inequality, identifying and drawing the boundary line is an essential step. The boundary line divides the plane into two regions, one of which contains all the solutions to the inequality.

• For inequalities such as \(y > mx + b\), or \(y < mx + b\), the boundary line is dashed.
• For \(y \geq mx + b\), or \(y \leq mx + b\), the line is solid, indicating that points on the line are included in the solution set.

To graph the boundary line, start by marking the y-intercept on the y-axis. Use the slope to find another point on the line: rise over run. This process leads us to a clear depiction of where the inequality changes from true to false.

Inequality Solution Region

The inequality solution region is the portion of the graph that satisfies the inequality. Once the boundary line is drawn, the next step is determining which side of the line contains the solutions.

• Choose a test point not on the boundary line, a common and easy choice is the origin, \((0,0)\), except when it lies on the line.
• Substitute the test point into the inequality. If the inequality holds true, the side containing the test point is the solution region.
• If false, shade the opposite side.

For the inequality \(y > mx + b\), solutions lie above the line if ascends, or one may need to check based on the specific coordinates and line direction. Shading this region visually represents all possible solutions to the inequality equation, creating a quick visual reference for understanding and validating solutions.