Problem 60 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x \geq 0 \\ y \geq 0 \\ 2 x+y<4 \\ 2 x-3 y \leq 6 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set of the given system of inequalities is the area that satisfies all four inequalities simultaneously. This area can be visualized on the graph as the region where the shading for all four inequalities overlaps.

Step 1: Graphing the inequation x ≥ 0

The inequality \( x \geq 0 \) describes all points to the right of the y-axis, including the y-axis itself. To graph this, draw a solid vertical line at x = 0, and shade the area to the right of this line.

Step 2: Graphing the inequation y ≥ 0

The inequality \( y \geq 0 \) specifies all the points above the x-axis, including the x-axis. This is represented by drawing a solid horizontal line at y = 0 and shading the region above this line.

Step 3: Graphing the inequation 2x + y < 4

To graph the line \( 2x+y = 4 \), firstly convert it to slope-intercept form by isolating y: \( y = -2x + 4 \). This line has a y-intercept of 4 and a slope of -2. Thus, begin at the point (0, 4) and move two units down and one unit to the right. Since the original inequality uses a less than symbol, we will use a dashed line to represent the boundary line and shade the region beneath it because the original inequality does not include the boundary line itself.

Step 4: Graphing the inequation 2x - 3y ≤ 6

First, let's rearrange \( 2x-3y=6 \) into slope-intercept form to graph it: \( y = \frac{2}{3}x - 2 \). This line's y-intercept is -2 and its slope is \( \frac{2}{3} \). So, start at the point (0, -2) and move two units up and three units to the right. As the original inequality includes the boundary line (represented by the less than or equal to symbol), draw a solid line and shade the region beneath this line.

Step 5: Identifying the Solution Set

The solution set is the region where the shading for all four inequalities overlaps. This is a region that satisfies all four inequality conditions at the same time.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing solutions of inequalities on a coordinate plane. Each inequality in a system is graphed one at a time.
We start by converting the inequality equations into lines. If the inequality symbol is ≥ or ≤, the line should be solid, indicating that points on the line satisfy the inequality.
If the inequality symbol is > or <, use a dashed line, as points on the line aren't part of the solution.

• Identify the boundary line by treating each inequality as an equation and solving for y in terms of x, if needed.
• Choose a test point not on the boundary (often the origin) to determine which side of the line to shade.
• Shade the area representing the solution to the inequality. Repeat for each inequality.

The final solution of the system is the overlapping shaded region on the graph.

Solution Set

The solution set of a system of inequalities is the set of all points that satisfy all inequalities simultaneously. After graphing each inequality, you're left with shaded regions. The intersection of these regions is the solution set.
It visually represents all possible solutions that fit the conditions of every inequality in the system.
Finding the solution set requires precision:

• Ensure that you accurately draw each boundary line, using either solid or dashed lines based on the inequality's requirements.
• The solution set must fall inside the combined shaded areas of all the inequalities.
• Any point within this shared region will satisfy all the inequalities.

Understanding where all four inequalities overlap is crucial in deciding the solution set of multiple inequalities.

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations as y = mx + c, where m is the slope, and c is the y-intercept.
This form simplifies graphing processes by allowing you to quickly identify the starting point and direction of a line.

• Start by plotting the y-intercept on the graph, which is the point where the line crosses the y-axis (0, c).
• The slope 'm' describes the line's angle on the graph, calculated as 'rise over run'. Move vertically by the rise and horizontally by the run from the y-intercept to draw the line.
• Slope-intercept form is particularly useful when converting standard form equations into a more graph-friendly version.

Using \[-\frac{2}{3}\] and \[\frac{2}{3}\] as examples for different slopes can help visualize how lines tilt on a graph.

Inequality Graph Shading

Shading on inequality graphs visually highlights the regions that satisfy the inequality. After graphing the boundary line, shading indicates where potential solutions exist.
The type of shading depends on the inequality symbol, and understanding this is key to correctly representing solutions.

• For inequalities with > or ≥, the shading is above the line when in slope-intercept form, as all y values satisfy the inequality for points above.
• For < or ≤, shade below the line, since the y values are smaller, satisfying the inequality.
• Overlap in shaded areas for multiple inequalities shows where all conditions are satisfied, forming the system's solution set.

The shade technique gives a visual reference, making it easier to identify correct solutions and understand relationships between inequalities.