Problem 31 - Introductory Algebra for College Students

Question

Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{l}4 x-3 y>12 \\\x \geq 0 \\\y \leq 0\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set lies in the quadrant IV, and limited by the line y = -4/3x + 4 on its right/top side.

Step 1 - Convert to y = mx + b

First, convert \[4x -3y > 12\] to \[y < -\frac{4}{3}x + 4\], \[x \geq 0 \] stays as it is, and \[y \leq 0\] stays as it is too.

Step 2 - Graph each inequality-individual line

Start by graphing each line one at a time. Begin with the line \[y = -\frac{4}{3}x + 4\] which is a line with a slope of -4/3 and y-intercept of 4. Then graph the line x=0 which is the y-axis. Finally, graph the line y=0, which is the x-axis.

Step 3 - Shade the correct half-plane

In general, for inequality \'greater than\' the area above the line is shaded and for \'less than\' the area below the line is shaded. For \[y < -\frac{4}{3}x + 4\], shade below the line. For \[x \geq 0\], shade to the right of the y-axis. For \[y \leq 0\], shade below the x-axis.

Step 4 - Identify the solution set

The solution set is the region that is shaded for all inequalities. It is the intersection of all the shaded regions.

Key Concepts

Linear Inequalities

Linear inequalities are similar to linear equations, but instead of using an equal sign, they use inequality symbols like ">", "<", "≥", and "≤". These inequalities represent a range of values rather than a single point solution. Understanding linear inequalities involves recognizing the relationships between variables. For the inequality \(4x - 3y > 12\), it can be rewritten in slope-intercept form as \(y < -\frac{4}{3}x + 4\), meaning we're looking for all points \((x,y)\) that lie below the line \(y = -\frac{4}{3}x + 4\). Key elements to check with linear inequalities include:

• Inequality direction: Determines whether we shade above or below the line.
• Boundary lines: These are graphed lines where the inequality becomes an equation.

By understanding these components, you can solve and visualize inequalities accurately.

Graphing Inequalities

Graphing inequalities is a method to visually represent their solutions. It involves several key steps to accurately display the regions that satisfy the given inequalities. Start with each inequality separately and convert it to an equation for a clearer boundary line. For instance, with \( y < -\frac{4}{3}x + 4 \), the line \( y = -\frac{4}{3}x + 4 \) acts as a boundary. To graph:

• Draw the boundary line. For "<" or ">", use a dashed line, indicating the points on the line are not included in the solution.
• For "≤" or "≥", use a solid line as the points on the line are included.
• Slope determines the line's angle and intercepts the point it crosses the y-axis.

Once individual lines are drawn, the focus shifts to the inequalities they represent.

Solution Set

The solution set of a system of inequalities is the region where all the specified conditions overlap. If you have multiple inequalities, as in this example, the solution set is found where all shaded areas from individual inequalities intersect.For our example, the solution set is where:

• Points below the line \( y = -\frac{4}{3}x + 4 \) coincide with
• Points to the right of the y-axis \( x \geq 0 \)
• And points below the x-axis \( y \leq 0 \)

The overlapping region fulfills all three conditions simultaneously. Identifying this area provides a visual representation of all possible solutions to the system.

Coordinate Plane

The coordinate plane is essential for graphing and solving systems of inequalities. It's a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Together, they create a grid used to plot points, lines, and regions.Understanding the coordinate plane:

• Each point is defined by an \( (x, y) \) coordinate.
• Quadrants divide the plane, with the origin (0,0) at the center.
• The axes themselves act as boundaries—for this system, lines \( x = 0 \) (y-axis) and \( y = 0 \) (x-axis) are pivotal.

In the context of our exercise, the plane helps identify where all parts of the solution set lie relative to these axes and other graphed inequalities.