Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the graphical solution of each inequality. $$3 x+4 y \leq-2$$

Step-by-Step Solution

Answer

The graphical solution of the inequality \(3x + 4y \leq -2\) is the region below the line passing through the points \((- \frac{2}{3}, 0)\) and \((0, -\frac{1}{2})\).

Step 1: Convert the inequality into a line equation

Take the inequality: \(3x + 4y \leq -2\) Replace the inequality sign with an equal sign to get the line equation: \(3x + 4y = -2\)

Step 2: Graph the line on a coordinate plane

To graph the line equation, we need to find two points that are on the line. We can do this by solving for x and y. When \(x = 0\), solve for \(y\): \(3(0) + 4y = -2\) \(4y = -2\) \(y = -\frac{1}{2}\) When \(y = 0\), solve for \(x\): \(3x + 4(0) = -2\) \(3x = -2\) \(x = -\frac{2}{3}\) These two points are \((- \frac{2}{3}, 0)\) and \((0, -\frac{1}{2})\). Plot these points on the coordinate plane and draw the line through them.

Step 3: Identify the region that satisfies the inequality

Now, we must determine which region on the coordinate plane, above or below the line, satisfies the inequality. To do this, we can choose a test point that is not on the line and plug it into the original inequality. If the inequality is true, then the region that contains the test point is the solution. A common test point to use is the origin \((0, 0)\). Plug this point into the original inequality: \(3(0) + 4(0) \leq -2\) \(0 \leq -2\) The inequality is false, so the region that contains the origin is not the solution. Therefore, the region opposite the origin (below the line) is the solution. Now, sketch the region below the line on the graph. The graphical solution of the inequality \(3x + 4y \leq -2\) is the region below the line passing through the points \((- \frac{2}{3}, 0)\) and \((0, -\frac{1}{2})\).

Key Concepts

Coordinate Plane

The coordinate plane, sometimes known as the Cartesian plane, is a two-dimensional surface where we can graphically represent equations and inequalities. It is made up of two perpendicular number lines: the x-axis, which is horizontal, and the y-axis, which is vertical. These axes intersect at a point called the origin, defined by the coordinates

• (0, 0).

Every point on the coordinate plane is specified using a pair of numbers known as coordinates, written in the form

• (x, y).

The coordinate plane is divided into four quadrants based on the signs of the coordinates. Starting from the top right and moving counter-clockwise:

• Quadrant I (+, +),
• Quadrant II (-, +),
• Quadrant III (-, -),
• Quadrant IV (+, -).

Understanding the layout of the coordinate plane is essential for plotting points and drawing graphs to visually represent mathematical solutions.

Plotting Points

Plotting points on a coordinate plane is a fundamental skill needed to understand patterns, equations, and inequalities. Each point is determined by an ordered pair

• (x, y),

where 'x' indicates horizontal placement and 'y' indicates vertical placement. To plot a point:

1. Start at the origin
   • (0, 0).
2. Move horizontally to the x-coordinate value.
3. Move vertically to the y-coordinate value and mark the point.

For instance, to plot the point

• (- \frac{2}{3}, 0)

start at the origin, move \(- \frac{2}{3}\) to the left on the x-axis, and then 0 on the y-axis, since it sits directly on the x-axis. Similarly, for

• (0, - \frac{1}{2}),

move to the origin, stay there for x, and go \(- \frac{1}{2}\) downward on the y-axis. With practice, identifying these movements becomes intuitive and enables you to graph lines with ease.

Linear Inequalities

Linear inequalities are expressions that can describe relationships of quantities where one side is not equal to the other and can be less than or greater than. The inequality

• \(3x + 4y \leq -2\)

means that the collection of points (x, y) must satisfy this condition. To solve graphically:

1. Convert the inequality to an equation by replacing \(\leq\) with =.
2. Graph the resulting line, which acts as a boundary.
3. Determine which side of the line contains solutions to the inequality.

Use a test point not on the line (commonly

• (0, 0)

if it's not on the line) to check which region satisfies the inequality. Substitute this test point into the original inequality. If true, shade the region containing the test point. Otherwise, shade the opposite region. This shaded region graphically represents all possible solutions to the inequality, by showing which parts of the plane satisfy the condition \(3x + 4y \leq -2\).