Solve The Following System Of Inequalities Graphically On The Set Of Axes Below. State The Coordinates Of A Point In The Solution Set.${ \begin{array}{c} y \leq X + 6 \ y \geq -\frac{1}{2}x - 3 \end{array} }$

Mar 13, 2025 by ADMIN 210 views

**Solving a System of Inequalities Graphically**

Introduction

In this article, we will explore the concept of solving a system of inequalities graphically. A system of inequalities is a set of two or more inequalities that are combined to form a single solution set. In this case, we have two inequalities: $y \leq x + 6$ and $y \geq -\frac{1}{2}x - 3$ . Our goal is to find the solution set of this system of inequalities and state the coordinates of a point in the solution set.

Understanding the Inequalities

Before we can solve the system of inequalities, we need to understand the individual inequalities. The first inequality is $y \leq x + 6$ . This means that the value of $y$ is less than or equal to the value of $x$ plus 6. The second inequality is $y \geq -\frac{1}{2}x - 3$ . This means that the value of $y$ is greater than or equal to the value of $-\frac{1}{2}x$ minus 3.

Graphing the Inequalities

To solve the system of inequalities, we need to graph the individual inequalities on a set of axes. The first inequality, $y \leq x + 6$ , can be graphed by drawing a line with a slope of 1 and a y-intercept of 6. The inequality $y \leq x + 6$ is a solid line, indicating that the solution set includes the line itself.

**Graph of y ≤ x + 6**
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The line has a slope of 1.
The y-intercept is 6.
The line is solid, indicating that the solution set includes the line itself.

The second inequality, $y \geq -\frac{1}{2}x - 3$ , can be graphed by drawing a line with a slope of $-\frac{1}{2}$ and a y-intercept of -3. The inequality $y \geq -\frac{1}{2}x - 3$ is a solid line, indicating that the solution set includes the line itself.

**Graph of y ≥ -1/2x - 3**
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The line has a slope of -1/2.
The y-intercept is -3.
The line is solid, indicating that the solution set includes the line itself.

Finding the Solution Set

To find the solution set of the system of inequalities, we need to find the region where the two inequalities overlap. This can be done by shading the region between the two lines.

**Solution Set**
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The solution set is the region where the two inequalities overlap.
The solution set is shaded in the graph below.

Graph of the Solution Set

The graph below shows the solution set of the system of inequalities.

**Graph of the Solution Set**
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The solution set is shaded in the graph.
The solution set includes the region between the two lines.

Finding a Point in the Solution Set

To find a point in the solution set, we can choose a point that lies within the shaded region. One such point is (0, 3).

**Point in the Solution Set**
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The point (0, 3) lies within the shaded region.
The point (0, 3) is a point in the solution set.

Conclusion

In this article, we solved a system of inequalities graphically. We graphed the individual inequalities on a set of axes and found the solution set by shading the region between the two lines. We also found a point in the solution set by choosing a point that lies within the shaded region. The solution set of the system of inequalities is the region where the two inequalities overlap, and a point in the solution set is (0, 3).

Key Takeaways

A system of inequalities is a set of two or more inequalities that are combined to form a single solution set.
To solve a system of inequalities graphically, we need to graph the individual inequalities on a set of axes.
The solution set of a system of inequalities is the region where the two inequalities overlap.
A point in the solution set is a point that lies within the shaded region.

Final Thoughts

Introduction

In our previous article, we explored the concept of solving a system of inequalities graphically. We graphed the individual inequalities on a set of axes and found the solution set by shading the region between the two lines. In this article, we will answer some common questions related to solving a system of inequalities graphically.

Q: What is a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that are combined to form a single solution set.

Q: How do I graph a system of inequalities?

A: To graph a system of inequalities, you need to graph the individual inequalities on a set of axes. You can use a graphing calculator or draw the lines by hand.

Q: What is the solution set of a system of inequalities?

A: The solution set of a system of inequalities is the region where the two inequalities overlap.

Q: How do I find a point in the solution set?

A: To find a point in the solution set, you can choose a point that lies within the shaded region.

Q: What if the two inequalities are parallel?

A: If the two inequalities are parallel, they will not intersect and the solution set will be empty.

Q: What if the two inequalities are perpendicular?

A: If the two inequalities are perpendicular, they will intersect at a single point and the solution set will be a single point.

Q: Can I use a graphing calculator to solve a system of inequalities?

A: Yes, you can use a graphing calculator to solve a system of inequalities. Graphing calculators can graph the individual inequalities and find the solution set.

Q: What are some common mistakes to avoid when solving a system of inequalities?

A: Some common mistakes to avoid when solving a system of inequalities include:

Graphing the inequalities incorrectly
Not shading the region between the two lines
Not choosing a point that lies within the shaded region

Q: How can I apply solving a system of inequalities to real-world problems?

A: Solving a system of inequalities can be applied to a variety of real-world problems, such as:

Finding the region where a company's profits are greater than its costs
Determining the region where a product's price is less than its value
Finding the region where a company's revenue is greater than its expenses

Conclusion

Solving a system of inequalities graphically is a useful tool for finding the solution set of a system of inequalities. By graphing the individual inequalities on a set of axes and shading the region between the two lines, we can find the solution set and a point in the solution set. This is a useful skill to have in mathematics and can be applied to a variety of problems.

Key Takeaways

A system of inequalities is a set of two or more inequalities that are combined to form a single solution set.
To solve a system of inequalities graphically, you need to graph the individual inequalities on a set of axes.
The solution set of a system of inequalities is the region where the two inequalities overlap.
A point in the solution set is a point that lies within the shaded region.