Solve The Inequality And Graph The Solution.$\[ 18 \ \textgreater \ 33z - 48 \\]To Graph The Solution, Plot An Endpoint And Select An Arrow. Select An Endpoint To Change It From Closed To Open. Select The Middle Of The Ray To Delete It.

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Introduction

Inequalities are mathematical expressions that compare two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving and graphing inequalities are essential skills in mathematics, particularly in algebra and geometry. In this article, we will focus on solving and graphing linear inequalities, with a specific example of the inequality 18 > 33z - 48.

Understanding Linear Inequalities

A linear inequality is an inequality that can be written in the form ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c, where a, b, and c are constants, and x is the variable. To solve a linear inequality, we need to isolate the variable x on one side of the inequality sign.

Solving the Inequality 18 > 33z - 48

To solve the inequality 18 > 33z - 48, we need to isolate the variable z on one side of the inequality sign. We can do this by adding 48 to both sides of the inequality:

18 > 33z - 48 18 + 48 > 33z - 48 + 48 66 > 33z

Next, we can divide both sides of the inequality by 33 to solve for z:

66/33 > 33z/33 2 > z

Therefore, the solution to the inequality 18 > 33z - 48 is z < 2.

Graphing the Solution

To graph the solution to the inequality z < 2, we need to plot an endpoint and select an arrow. The endpoint is the value of z that makes the inequality true, which is z = 2. Since the inequality is less than (<), we need to select an open circle to indicate that the endpoint is not included in the solution.

Selecting an Endpoint

To select an endpoint, we need to click on the point where the ray intersects the number line. In this case, we need to click on the point where z = 2. Once we select the endpoint, we can change it from closed to open by clicking on the "Select endpoint" button.

Deleting a Ray

To delete a ray, we need to click on the middle of the ray. In this case, we need to click on the middle of the ray that represents the solution to the inequality z < 2. Once we delete the ray, the solution will be represented by an open circle at the endpoint z = 2.

Conclusion

Solving and graphing inequalities are essential skills in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, we can solve and graph linear inequalities, such as the inequality 18 > 33z - 48. Remember to isolate the variable on one side of the inequality sign, and to select an open circle to indicate that the endpoint is not included in the solution.

Tips and Tricks

  • When solving an inequality, make sure to isolate the variable on one side of the inequality sign.
  • When graphing an inequality, make sure to select an open circle to indicate that the endpoint is not included in the solution.
  • When deleting a ray, make sure to click on the middle of the ray to delete it.

Common Mistakes

  • Failing to isolate the variable on one side of the inequality sign.
  • Failing to select an open circle to indicate that the endpoint is not included in the solution.
  • Failing to delete a ray when it is no longer needed.

Real-World Applications

Solving and graphing inequalities have many real-world applications, such as:

  • Modeling population growth and decline.
  • Modeling the spread of diseases.
  • Modeling the behavior of physical systems, such as springs and pendulums.

Conclusion

Introduction

In our previous article, we discussed how to solve and graph linear inequalities, with a specific example of the inequality 18 > 33z - 48. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving and graphing inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols.

Q: What are the different types of inequalities?

A: There are four main types of inequalities:

  • Greater than (>): a > b
  • Less than (<): a < b
  • Greater than or equal to (≥): a ≥ b
  • Less than or equal to (≤): a ≤ b

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What is the difference between a closed circle and an open circle on a number line?

A: A closed circle represents a value that is included in the solution, while an open circle represents a value that is not included in the solution.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to plot an endpoint and select an arrow. The endpoint is the value of the variable that makes the inequality true, and the arrow indicates the direction of the inequality.

Q: What is the significance of the endpoint in graphing an inequality?

A: The endpoint represents the value of the variable that makes the inequality true. If the inequality is greater than or equal to (≥), the endpoint is included in the solution. If the inequality is less than or equal to (≤), the endpoint is included in the solution. If the inequality is greater than (>), the endpoint is not included in the solution. If the inequality is less than (<), the endpoint is not included in the solution.

Q: How do I determine whether to use a closed circle or an open circle on a number line?

A: To determine whether to use a closed circle or an open circle on a number line, you need to look at the inequality sign. If the inequality sign is greater than or equal to (≥) or less than or equal to (≤), use a closed circle. If the inequality sign is greater than (>), use an open circle. If the inequality sign is less than (<), use an open circle.

Q: What are some common mistakes to avoid when solving and graphing inequalities?

A: Some common mistakes to avoid when solving and graphing inequalities include:

  • Failing to isolate the variable on one side of the inequality sign.
  • Failing to select the correct type of circle (closed or open) on a number line.
  • Failing to plot the endpoint correctly on a number line.
  • Failing to select the correct arrow on a number line.

Q: What are some real-world applications of solving and graphing inequalities?

A: Some real-world applications of solving and graphing inequalities include:

  • Modeling population growth and decline.
  • Modeling the spread of diseases.
  • Modeling the behavior of physical systems, such as springs and pendulums.
  • Determining the maximum or minimum value of a function.
  • Finding the intersection of two or more functions.

Conclusion

In conclusion, solving and graphing inequalities are essential skills in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can better understand the concepts and techniques involved in solving and graphing inequalities. Remember to isolate the variable on one side of the inequality sign, and to select the correct type of circle (closed or open) on a number line.