Problem 6 - Algebra for College Students

Question

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{l} 4 x+3 y \geq 12 \\ 3 x-4 y \geq 12 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution set is the region above the line \(y = -\frac{4}{3}x + 4\) and below the line \(y = \frac{3}{4}x - 3\), including the boundaries.

Step 1: Understand the Inequalities

We have the system of inequalities: \( 4x + 3y \geq 12 \) and \( 3x - 4y \geq 12 \). We need to graph each inequality on the coordinate plane and determine the region that satisfies both inequalities.

Step 2: Graph the First Inequality

First, rewrite the inequality \(4x + 3y \geq 12\) in slope-intercept form: \(y \geq -\frac{4}{3}x + 4\). This line intersects the y-axis at (0, 4) and has a slope of \(-\frac{4}{3}\). Draw a solid line through these points because the inequality is \(\geq\). Shade above the line as the inequality is greater than.

Step 3: Graph the Second Inequality

Next, rewrite the inequality \(3x - 4y \geq 12\) in slope-intercept form: \(y \leq \frac{3}{4}x - 3\). This line intersects the y-axis at (0, -3) and has a slope of \(\frac{3}{4}\). Draw a solid line through these points because the inequality is \(\geq\). Shade below the line, as the inequality is less than.

Step 4: Identify the Solution Set

Find the region where the shadings from both inequalities overlap. This overlapping region is the solution set for the system of inequalities. Ensure to rely on the algebraic verification if necessary, especially if the graph is complex.

Key Concepts

Solution Set

The solution set of a system of inequalities is the region where all the inequalities in the system are satisfied simultaneously. When graphing these systems, you will notice that each inequality represents a boundary line on the coordinate plane.

• The region where these lines overlap is your solution set.
• Think of it as the intersection of multiple shaded areas on your graph.
• This set of points (solutions) makes all inequalities true.

To correctly determine the solution set, ensure that you understand how each inequality affects the solution space. Sometimes, you might need to validate your findings algebraically despite what the visual graph shows. This confirmation is crucial, especially in more complex situations.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It's defined by two axes:

• Horizontal: x-axis
• Vertical: y-axis

These axes intersect at the origin, which is the point (0, 0). Each point on this plane has a pair of coordinates (x, y) that show its position.
When graphing inequalities, you'll use this plane to draw boundary lines and shade areas that represent solutions. By utilizing the coordinate plane, you can visualize where the inequalities' solutions intersect, helping you find the solution set.

Slope-Intercept Form

Slope-intercept form is a way to express linear equations and is given by the formula: \[ y = mx + b \]Here's what this means:

• m: Slope of the line, telling you how steep the line is.
• b: Y-intercept, which shows where the line crosses the y-axis.

When graphing inequalities, converting them to slope-intercept form makes it easier to plot boundary lines. For example, in the equation \( y \geq -\frac{4}{3}x + 4 \), the slope \(-\frac{4}{3}\) indicates the line's tilt, and the y-intercept 4 tells you where it hits the y-axis. This form simplifies graphing and helps accurately identify areas to shade.

Systems of Inequalities

Systems of inequalities involve finding a solution that satisfies more than one inequality at the same time. These systems can be solved graphically by plotting each inequality on the coordinate plane.

• Each inequality will have its boundary line.
• The appropriate region for each inequality is shaded.
• The solution to the system is the overlapping area where all shaded regions intersect.

Converting each inequality to slope-intercept form aids in plotting these lines swiftly. Once graphed, it’s essential to check which side of the boundary to shade. Use test points if necessary for confirmation. Ultimately, the common area provides a visual representation of the solution set for the system of inequalities.