Let $z$ Be The Elevation Of A Passage In Meters. Write An Inequality That Represents Elevations Below -1.5 Meters.$z \ \textless \ -1.5$Graph The Inequality $ Z \textless − 1.5 Z \ \textless \ -1.5 Z \textless − 1.5 [/tex].

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Introduction

In mathematics, inequalities are used to represent relationships between variables. They are an essential part of mathematical expressions and are used to describe various conditions. In this article, we will focus on a specific inequality that represents elevations below a certain level. We will also explore how to graph this inequality.

The Inequality

Let $z$ be the elevation of a passage in meters. We want to write an inequality that represents elevations below -1.5 meters. The inequality we are looking for is:

z<−1.5z < -1.5

This inequality states that the elevation $z$ is less than -1.5 meters.

Graphing the Inequality

To graph the inequality $z < -1.5$, we need to understand that it represents all values of $z$ that are less than -1.5. This means that any value of $z$ that is less than -1.5 will satisfy the inequality.

Step 1: Draw a Number Line

To graph the inequality, we start by drawing a number line. A number line is a line that represents all possible values of a variable. In this case, the variable is $z$, which represents the elevation in meters.

Step 2: Mark the Value -1.5

Next, we mark the value -1.5 on the number line. This value represents the boundary of the inequality.

Step 3: Shade the Region

Since the inequality represents all values of $z$ that are less than -1.5, we shade the region to the left of -1.5 on the number line. This region represents all values of $z$ that satisfy the inequality.

The Graph

The graph of the inequality $z < -1.5$ is a number line with a shaded region to the left of -1.5. This graph represents all values of $z$ that are less than -1.5 meters.

Example

Suppose we want to find the elevation of a passage that is below -1.5 meters. We can use the graph of the inequality to find the solution. Since the graph shows that all values of $z$ to the left of -1.5 satisfy the inequality, we can conclude that any elevation below -1.5 meters will satisfy the inequality.

Conclusion

In this article, we have explored the inequality $z < -1.5$ and its graphical representation. We have seen how to graph the inequality and how to use it to find the solution to a problem. Inequalities are an essential part of mathematics, and understanding how to graph them is crucial for solving mathematical problems.

Key Takeaways

  • Inequalities are used to represent relationships between variables.
  • The inequality $z < -1.5$ represents all values of $z$ that are less than -1.5 meters.
  • To graph the inequality, we draw a number line and mark the value -1.5.
  • We shade the region to the left of -1.5 to represent all values of $z$ that satisfy the inequality.
  • The graph of the inequality is a number line with a shaded region to the left of -1.5.

Further Reading

For further reading on inequalities and their graphical representation, we recommend the following resources:

References

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two values or expressions using a relation such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤).

Q: How do I graph an inequality?

A: To graph an inequality, you need to draw a number line and mark the value or expression that is being compared. Then, you shade the region to the left or right of the marked value, depending on the direction of the inequality.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses a strict relation, such as < or >. A non-strict inequality is an inequality that uses a non-strict relation, such as ≤ or ≥.

Q: How do I determine the direction of the inequality?

A: To determine the direction of the inequality, you need to look at the relation used in the inequality. If the relation is < or >, the inequality is strict and the region to the left or right of the marked value is shaded. If the relation is ≤ or ≥, the inequality is non-strict and the region to the left or right of the marked value is shaded, including the marked value itself.

Q: Can I have multiple inequalities on the same graph?

A: Yes, you can have multiple inequalities on the same graph. To do this, you need to draw a number line and mark the values or expressions that are being compared. Then, you shade the regions to the left or right of the marked values, depending on the direction of the inequalities.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the values that satisfy all the inequalities in the system. You can do this by graphing the inequalities on the same number line and finding the region where all the inequalities overlap.

Q: What is the importance of graphing inequalities?

A: Graphing inequalities is an important skill in mathematics because it helps you visualize the relationships between variables and understand the conditions that must be met. It is also a useful tool for solving problems and making decisions.

Q: Can I use graphing inequalities in real-life situations?

A: Yes, you can use graphing inequalities in real-life situations. For example, you can use it to compare the prices of different products, to determine the best investment option, or to find the optimal solution to a problem.

Q: How do I practice graphing inequalities?

A: You can practice graphing inequalities by working through examples and exercises in a textbook or online resource. You can also try graphing inequalities on your own using a number line and marking the values or expressions that are being compared.

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

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

  • Not marking the value or expression that is being compared
  • Not shading the correct region
  • Not considering the direction of the inequality
  • Not solving the system of inequalities correctly

Q: How do I know if I am graphing inequalities correctly?

A: To know if you are graphing inequalities correctly, you need to check your work by:

  • Verifying that you have marked the value or expression that is being compared
  • Checking that you have shaded the correct region
  • Ensuring that you have considered the direction of the inequality
  • Solving the system of inequalities correctly

Conclusion

Graphing inequalities is an important skill in mathematics that helps you visualize the relationships between variables and understand the conditions that must be met. By practicing graphing inequalities, you can improve your problem-solving skills and make informed decisions in real-life situations.