Problem 44 - Introductory Algebra for College Students

Question

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}y \geq-3 x+2 \\\y<-3 x \\\x \geq 1\end{array}\right.$$

Step-by-Step Solution

Answer

The system of linear inequalities has no solutions because there's no region in the coordinate plane that satisfies all three inequalities simultaneously.

Step 1: Discuss the inequalities individually

The exercise provided three inequalities: \(y \geq -3x + 2\), \(y < -3x\), and \(x \geq 1\). The first inequality represents a line with a slope of -3 and a y-intercept at 2, and all the solutions are above and on the line. The second inequality represents a line with slope -3 without a y-intercept (passing through the origin), and all the solutions lie under the line. The third inequality represents a vertical line at x=1, and all the solutions lie on and to the right of this line.

Step 2: Graph the inequalities

Using the properties mentioned in Step 1, graph the three inequalities on the same coordinate plane. Remember to use a different line style for each inequality (solid for \(y \geq -3x + 2\), dotted for \(y < -3x\), and solid for \(x \geq 1\)). Don't forget to shade the regions indicating the solutions for each inequality (above and on the line for \(y \geq -3x + 2\), below the line for \(y < -3x\), and on and to the right of the line for \(x \geq 1\)).

Step 3: Identify the solution set

When you've constructed the graphs, look for the area where all shaded regions overlap. This is the solution set for the system of inequalities. However, in this case, there is no area where all three shaded regions overlap.

Step 4: Conclude

Since no region satisfies all three inequalities simultaneously, conclude that there is no solution to this system of inequalities and explain that this happens when the conditions of the inequalities make it impossible for any point (x, y) to fulfill all conditions simultaneously.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting regions on the coordinate plane that satisfy the inequality. Each inequality in the exercise corresponds to a line, and the solutions lie either above, below, or on this line.

• For the inequality \(y \geq -3x + 2\), plot a solid line because it includes points on the line itself. The region above this line represents solutions.
• The inequality \(y < -3x\) requires a dashed line. Points are not on the line itself, so you shade the area below it to represent the solutions.
• For \(x \geq 1\), draw a solid vertical line at \(x = 1\), with the shading extending to the right, indicating all x-values greater than or equal to 1.

Using consistent line styles and shading helps clearly identify the regions that satisfy each inequality. Carefully graph each line and shade the correct region to visually find where solutions overlap.

Solution Set

In the context of inequalities, the solution set is the set of all points that satisfy all the inequalities in the system.
For the three given inequalities:

• Find the region where all shaded areas overlap when graphed. This overlap represents the solution set of the system of inequalities.
• If any inequality does not overlap with the others, no solution exists. This means no single point can satisfy all equations simultaneously.

In this exercise, the solution set doesn’t exist because the shaded regions don’t overlap in one common area.
This lack of overlap indicates that the conditions imposed by the inequalities cannot be fulfilled by any point \((x,y)\) in the coordinate plane.

Systems of Inequalities

Systems of inequalities involve multiple inequalities working together to define a particular region of solutions. Graphing each inequality and identifying overlaps of shaded regions are fundamental aspects of solving such systems.

• This exercise included three inequalities: some are complex and contrast sharply, making visualizing crucial for understanding their interactions.
• Sometimes, as with this exercise, no common region exists that satisfies all inequalities, meaning no collective solution is possible.
• Understanding how different inequalities intersect or don’t interact is crucial in comprehending systems of inequalities.

By practicing the graphing technique, you gain invaluable skills for visually analyzing and interpreting multiple conditions. Recognizing such patterns and their implications helps when approaching more advanced mathematical tasks.