Identifying the Solution Set

Oct 13, 2025 by recoveryshake.com 115 views

Introduction

In mathematics, inequalities are a fundamental concept that helps us compare values and make decisions. When dealing with linear inequalities, we often need to find the solution set that satisfies both inequalities. In this article, we will explore two linear inequalities and their corresponding graphs. By analyzing the graphs and using our understanding of inequalities, we will identify the point that satisfies both inequalities.

The Inequalities

We are given two linear inequalities:

y > -\frac{3}{2}x + 5
y < 2x - 3

Graphing the Lines

To visualize the inequalities, we need to graph the corresponding lines. The first inequality, y > -\frac{3}{2}x + 5, has a slope of -\frac{3}{2} and a y-intercept of 5. The second inequality, y < 2x - 3, has a slope of 2 and a y-intercept of -3.

### Graph 1: y > -\frac{3}{2}x + 5

Slope: -\frac{3}{2}
Y-intercept: 5

### Graph 2: y < 2x - 3

* Slope: 2
* Y-intercept: -3

Analyzing the Graphs

By examining the graphs, we can see that the first line has a negative slope, indicating that it slopes downward from left to right. The second line has a positive slope, indicating that it slopes upward from left to right.

### Graph 1: y > -\frac{3}{2}x + 5

The line slopes downward from left to right.
The y-intercept is 5, indicating that the line crosses the y-axis at (0, 5).

### Graph 2: y < 2x - 3

* The line slopes upward from left to right.
* The y-intercept is -3, indicating that the line crosses the y-axis at (0, -3).

Identifying the Solution Set

To find the point that satisfies both inequalities, we need to find the region where the two lines intersect. Since the first line has a negative slope and the second line has a positive slope, the two lines will intersect at some point.

### Intersection Point

The intersection point is the solution set that satisfies both inequalities.
To find the intersection point, we need to solve the system of equations formed by the two inequalities.

Solving the System of Equations

To solve the system of equations, we can set the two equations equal to each other and solve for x.

### Solving for x

-\frac{3}{2}x + 5 = 2x - 3
-\frac{5}{2}x = -8
x = \frac{16}{5}

Finding the Corresponding y-Value

Now that we have found the x-coordinate of the intersection point, we can substitute it into one of the original equations to find the corresponding y-value.

### Finding the y-Value

y > -\frac{3}{2}x + 5
y > -\frac{3}{2}(\frac{16}{5}) + 5
y > \frac{24}{5}

Conclusion

In conclusion, we have identified the point that satisfies both inequalities by analyzing the graphs and solving the system of equations. The intersection point is (x, y) = (\frac{16}{5}, \frac{24}{5}). This point satisfies both inequalities and lies in the region where the two lines intersect.

Final Answer

Q&A: Solving Inequalities

Q: What is the main difference between solving linear equations and solving linear inequalities? A: The main difference between solving linear equations and solving linear inequalities is that inequalities have a "greater than" or "less than" symbol, which indicates that the solution set is a range of values rather than a single value.

Q: How do I graph a linear inequality? A: To graph a linear inequality, you need to graph the corresponding line and then shade the region that satisfies the inequality. If the inequality is "greater than" or "less than," you will shade the region above or below the line, respectively.

Q: What is the significance of the y-intercept in a linear inequality? A: The y-intercept in a linear inequality is the point where the line crosses the y-axis. This point is important because it helps you determine the direction of the inequality.

Q: How do I find the intersection point of two linear inequalities? A: To find the intersection point of two linear inequalities, you need to solve the system of equations formed by the two inequalities. This can be done by setting the two equations equal to each other and solving for x.

Q: What is the solution set of a linear inequality? A: The solution set of a linear inequality is the set of all points that satisfy the inequality. This can be represented graphically as a region on the coordinate plane.

Q: Can a linear inequality have multiple solution sets? A: Yes, a linear inequality can have multiple solution sets. This can occur when the inequality has multiple lines or when the inequality is a combination of multiple inequalities.

Q: How do I determine the direction of a linear inequality? A: To determine the direction of a linear inequality, you need to examine the slope of the line and the y-intercept. If the slope is positive, the inequality is "greater than" or "less than." If the slope is negative, the inequality is "less than" or "greater than."

Q: Can a linear inequality have a single solution? A: Yes, a linear inequality can have a single solution. This can occur when the inequality has a single line and the solution set is a single point.

Q: How do I represent a linear inequality graphically? A: To represent a linear inequality graphically, you need to graph the corresponding line and then shade the region that satisfies the inequality. If the inequality is "greater than" or "less than," you will shade the region above or below the line, respectively.

Q: What is the significance of the slope in a linear inequality? A: The slope in a linear inequality is the rate of change of the line. This helps you determine the direction of the inequality and the shape of the solution set.

Q: Can a linear inequality have a solution set that is a single point? A: Yes, a linear inequality can have a solution set that is a single point. This can occur when the inequality has a single line and the solution set is a single point.

Q: How do I determine if a point satisfies a linear inequality? A: To determine if a point satisfies a linear inequality, you need to substitute the x and y values of the point into the inequality and check if the inequality is true.

Q: Can a linear inequality have a solution set that is a line? A: Yes, a linear inequality can have a solution set that is a line. This can occur when the inequality has a single line and the solution set is a line.

Q: How do I represent a linear inequality as a system of equations? A: To represent a linear inequality as a system of equations, you need to rewrite the inequality as a pair of equations and then solve the system of equations.

Q: What is the significance of the y-axis in a linear inequality? A: The y-axis in a linear inequality is the vertical line that passes through the origin. This line helps you determine the direction of the inequality and the shape of the solution set.

Q: Can a linear inequality have a solution set that is a region? A: Yes, a linear inequality can have a solution set that is a region. This can occur when the inequality has multiple lines and the solution set is a region.

Q: How do I determine the direction of a linear inequality using the slope? A: To determine the direction of a linear inequality using the slope, you need to examine the slope of the line. If the slope is positive, the inequality is "greater than" or "less than." If the slope is negative, the inequality is "less than" or "greater than."