Problem 30 - Intermediate Algebra

Question

Graph the solution set of each system of inequalities on a rectangular coordinate system. $$\left\\{\begin{array}{l}x+2 y<3 \\\2 x+4 y<8\end{array}\right.$$

Step-by-Step Solution

Answer

Graph the lines \(y = -\frac{1}{2}x + \frac{3}{2}\) and \(y = -\frac{1}{2}x + 2\), shading below both, the overlap is the solution.

Step 1: Understand the Inequalities

The problem gives two inequalities: \(x + 2y < 3\) and \(2x + 4y < 8\). We need to convert each of these into a form that's easy to graph on a coordinate plane to find their intersection.

Step 2: Rewrite as Equations

Rewrite the inequalities as equalities to find the boundary lines: \(x + 2y = 3\) and \(2x + 4y = 8\). These lines are the boundaries of the regions that satisfy the inequalities.

Step 3: Calculate the Slope-Intercept Form

Convert each equation to the slope-intercept form \(y = mx + b\). For \(x + 2y = 3\), rearrange to get \(y = -\frac{1}{2}x + \frac{3}{2}\). For \(2x + 4y = 8\), simplify to get \(y = -\frac{1}{2}x + 2\).

Step 4: Graph the Boundary Lines

Plot the boundary lines \(y = -\frac{1}{2}x + \frac{3}{2}\) and \(y = -\frac{1}{2}x + 2\) on the coordinate plane. Use dashed lines, as the inequalities are not inclusive (\(<\) not \(\leq\)).

Step 5: Shade the Solution Regions

For \(x + 2y < 3\), shade below the line since we want values less than the line. For \(2x + 4y < 8\), shade below its line too. The solution set is where the shaded regions overlap.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most valuable tools in algebra when working with lines on a graph. It is expressed as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form makes it straightforward to identify how the line behaves on the coordinate plane.

- **Slope (\( m \))**: This indicates the tilt or steepness of the line. It is calculated as the change in y divided by the change in x (rise over run). For example, in the equation \( y = -\frac{1}{2}x + \frac{3}{2} \), the slope \( m \) is \(-\frac{1}{2}\), showing a negative slope, meaning the line moves downward from left to right.
- **Y-intercept (\( b \))**: This is where the line crosses the y-axis. Using the same example, the y-intercept is \( \frac{3}{2} \), showing the point (0, \( \frac{3}{2} \)).

By converting equations to this form, you can easily graph them, and it's especially useful when solving systems of equations or inequalities on a coordinate plane.

Coordinate Plane

A coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It is a fundamental tool for graphing and visualizing mathematical relationships.

- **Axes**: The point where the x-axis and y-axis intersect is called the origin, marked as (0,0).
- **Quadrants**: The plane is divided into four quadrants based on the axes. The top right is the first quadrant, and the numbering follows a counterclockwise direction.

To graph any equation or inequality:

• Identify important points, such as intercepts and corners.
• Plot these points on the plane.
• Draw the line, using the slope to determine direction.

When working with inequalities, dashed or solid lines indicate whether the boundary is part of the solution set.

Solution Set

The solution set for a system of inequalities represents all the coordinates on the graph that satisfy all the given inequalities. It is the overlap of the shaded regions on the graph.

- **Graphical Representation**: When solving, each inequality is graphed on the same coordinate plane. The regions that satisfy the inequalities are shaded. For example, with the inequalities \( x+2y<3 \) and \( 2x+4y<8 \), you shade below both lines as they are strict inequalities (< not ≤).
- **Intersection of Shaded Regions**: The part of the graph where the shaded areas overlap represents the solution set. This is the region that satisfies all inequalities in the system simultaneously.

In practice, clearly identifying and shading overlapping regions helps visually determine the range of possible solutions.