Graphing And Solving Systems Of Linear Inequalities
In the realm of mathematics, understanding inequalities and their graphical representations is a fundamental skill. When we delve into the world of linear inequalities, we encounter situations where the relationship between variables isn't defined by a strict equality but rather by a range of possible values. This exploration focuses on dissecting the system of inequalities $y \leq 3x + 1$ and $y \geq -x + 2$, aiming to decipher the characteristics of their graphs and identify the solutions that satisfy both conditions.
Understanding Linear Inequalities
Before we dive into the specifics of our given system, let's first establish a firm understanding of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). When graphed on a coordinate plane, a linear inequality represents a region, rather than a single line, encompassing all the points that satisfy the inequality.
The boundary line of an inequality is the line that separates the region of solutions from the region of non-solutions. This line is determined by replacing the inequality symbol with an equal sign and graphing the resulting linear equation. The nature of the boundary line – whether it's solid or dashed – depends on the inequality symbol used. A solid line indicates that the points on the line are included in the solution set (≤ or ≥), while a dashed line signifies that the points on the line are not part of the solution set (< or >).
Analyzing the Given System of Inequalities
Our focus is on the system of inequalities:
To fully grasp the graphical representation and solution set of this system, we need to analyze each inequality individually and then consider their combined effect.
Inequality 1: $y \leq 3x + 1$
Let's dissect the first inequality, $y \leq 3x + 1$. To graph this inequality, we first treat it as an equation, $y = 3x + 1$, which represents a straight line. This line is the boundary line for our inequality. The equation is in slope-intercept form, y = mx + c, where m represents the slope and c represents the y-intercept. For the line $y = 3x + 1$, the slope (m) is 3, and the y-intercept (c) is 1. This means the line crosses the y-axis at the point (0, 1), and for every one unit increase in x, y increases by 3 units.
The boundary line in this case is a solid line. This is because the inequality includes an "equal to" component (≤), meaning all points on the line are included in the solution. A dashed line would be used if the inequality was strict (<).
Now, consider the inequality $y \leq 3x + 1$. This indicates we are interested in all the points where the y-coordinate is less than or equal to the value of 3x + 1. Graphically, this corresponds to the region below the line $y = 3x + 1$. To visualize this, you can choose a test point that is not on the line, such as (0, 0), and substitute it into the inequality. If the resulting statement is true, the region containing the test point is the solution region. If it’s false, the other region is the solution. Substituting (0, 0) into $y \leq 3x + 1$ gives us $0 \leq 3(0) + 1$, which simplifies to $0 \leq 1$, a true statement. Therefore, we shade the region below the line, as it contains the points that satisfy the inequality.
Inequality 2: $y \geq -x + 2$
Next, we turn our attention to the second inequality, $y \geq -x + 2$. Similar to the first inequality, we begin by considering the boundary line, which is the line represented by the equation $y = -x + 2$. This equation is also in slope-intercept form, y = mx + c, allowing us to easily identify the slope and y-intercept. Here, the slope (m) is -1, and the y-intercept (c) is 2. This means the line intersects the y-axis at the point (0, 2), and for every one unit increase in x, y decreases by 1 unit.
As with the previous inequality, the boundary line is solid. This is because the inequality is inclusive (≥), meaning that points on the line are included in the solution set. A dashed line would only be used if the inequality were strict (>).
The inequality $y \geq -x + 2$ describes all points where the y-coordinate is greater than or equal to the value of -x + 2. Graphically, this corresponds to the region above the line $y = -x + 2$. To determine which side of the line to shade, we can again use a test point. Let’s use (0, 0) again. Substituting into $y \geq -x + 2$ gives us $0 \geq -0 + 2$, which simplifies to $0 \geq 2$, a false statement. Thus, the region that contains (0, 0) is not the solution region, and we shade the region above the line $y = -x + 2$.
Graphical Solution of the System of Inequalities
Now that we've analyzed each inequality separately, we can combine our understanding to solve the system graphically. The solution to a system of inequalities is the region of the coordinate plane that satisfies all the inequalities simultaneously. This is the region where the shaded areas of the individual inequalities overlap.
To find this region, we overlay the graphs of the two inequalities. The region where the shading from both inequalities overlaps represents the solution set of the system. Any point within this overlapping region, or on the solid boundary lines, is a solution to the system of inequalities.
Evaluating the Given Statements
With a solid understanding of the graphical representation, we can now address the statements provided:
A. The slope of one boundary line is 2.
B. Both boundary lines are solid.
C. A solution to the system is (1,3).
Statement A: The slope of one boundary line is 2.
To evaluate this statement, we must recall the slopes of the boundary lines. The first inequality, $y \leq 3x + 1$, has a boundary line with a slope of 3, as deduced from its slope-intercept form. The second inequality, $y \geq -x + 2$, has a boundary line with a slope of -1. Neither boundary line has a slope of 2. Therefore, statement A is false.
Statement B: Both boundary lines are solid.
As we analyzed earlier, the boundary line for $y \leq 3x + 1$ is solid because the inequality includes the "equal to" component (≤). Similarly, the boundary line for $y \geq -x + 2$ is also solid because it includes the "equal to" component (≥). Thus, statement B is true. Solid lines indicate that the points on the lines are included in the solution set, which is the case for both inequalities here.
Statement C: A solution to the system is (1,3).
To determine if the point (1, 3) is a solution to the system, we need to substitute the coordinates x = 1 and y = 3 into both inequalities and check if they are satisfied.
For the first inequality, $y \leq 3x + 1$, we substitute x = 1 and y = 3:
This inequality holds true. Now, let's check the second inequality, $y \geq -x + 2$, with x = 1 and y = 3:
This inequality also holds true. Since the point (1, 3) satisfies both inequalities, statement C is true. The point (1, 3) lies within the region where the solutions of both inequalities overlap, making it a valid solution to the system.
Conclusion
In conclusion, by meticulously analyzing the system of inequalities $y \leq 3x + 1$ and $y \geq -x + 2$, we've determined that:
- The statement "The slope of one boundary line is 2" is false.
- The statement "Both boundary lines are solid" is true.
- The statement "A solution to the system is (1,3)" is true.
This exploration highlights the importance of understanding the fundamental concepts of linear inequalities, their graphical representations, and how to identify solutions that satisfy a system of such inequalities. By breaking down the problem into smaller parts—analyzing each inequality individually and then combining their solutions—we can effectively solve complex mathematical problems.
This comprehensive approach not only answers the specific question but also reinforces the broader understanding of mathematical concepts, empowering learners to tackle similar problems with confidence and precision.
