Write An Inequality For The Following Phrases:1. Three Less Than Twice A Number Is At Least -1.2. Four Less Than A Number Is More Than 7.

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Introduction

Inequalities are mathematical expressions that compare two values, often used to represent real-world situations. Writing inequalities from phrases is an essential skill in mathematics, as it allows us to model and solve problems in various fields. In this article, we will explore how to write inequalities for two given phrases.

Inequality 1: Three Less Than Twice a Number is at Least -1

The first phrase is "Three less than twice a number is at least -1." To write an inequality for this phrase, we need to identify the key elements:

  • Twice a number: This can be represented as 2x, where x is the unknown number.
  • Three less than: This means we need to subtract 3 from the previous expression, resulting in 2x - 3.
  • Is at least -1: This indicates that the expression 2x - 3 is greater than or equal to -1.

Using the above elements, we can write the inequality as:

2x - 3 ≥ -1

This inequality states that three less than twice a number is at least -1.

Inequality 2: Four Less Than a Number is More Than 7

The second phrase is "Four less than a number is more than 7." To write an inequality for this phrase, we need to identify the key elements:

  • Four less than a number: This can be represented as x - 4, where x is the unknown number.
  • Is more than 7: This indicates that the expression x - 4 is greater than 7.

Using the above elements, we can write the inequality as:

x - 4 > 7

This inequality states that four less than a number is more than 7.

Solving Inequalities

Inequalities can be solved using various methods, including algebraic manipulation and graphical representation. Here are some steps to solve the inequalities we wrote earlier:

Solving Inequality 1: 2x - 3 ≥ -1

To solve this inequality, we can add 3 to both sides, resulting in:

2x ≥ 2

Next, we can divide both sides by 2, resulting in:

x ≥ 1

This means that the solution to the inequality is x ≥ 1.

Solving Inequality 2: x - 4 > 7

To solve this inequality, we can add 4 to both sides, resulting in:

x > 11

This means that the solution to the inequality is x > 11.

Conclusion

Writing inequalities from phrases is an essential skill in mathematics, as it allows us to model and solve problems in various fields. In this article, we explored how to write inequalities for two given phrases and solved the resulting inequalities using algebraic manipulation. By following these steps, you can write and solve inequalities for a wide range of real-world situations.

Key Takeaways

  • Inequalities are mathematical expressions that compare two values.
  • Writing inequalities from phrases is an essential skill in mathematics.
  • Inequalities can be solved using algebraic manipulation and graphical representation.
  • The solution to an inequality is a range of values that satisfy the inequality.

Real-World Applications

Inequalities have numerous real-world applications, including:

  • Modeling population growth and decline
  • Representing financial transactions and budgets
  • Solving optimization problems in business and economics
  • Analyzing data and making predictions in science and engineering

Introduction

Writing inequalities from phrases is a fundamental skill in mathematics, and it's essential to understand the concepts and techniques involved. In this article, we'll address some frequently asked questions about writing inequalities, providing clear explanations and examples to help you master this skill.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often represented using symbols such as ≥, ≤, >, or <. Inequalities can be used to model real-world situations, such as comparing the cost of two products or the time it takes to complete a task.

Q: How do I write an inequality from a phrase?

A: To write an inequality from a phrase, follow these steps:

  1. Identify the key elements in the phrase, such as the unknown value, the operation, and the comparison.
  2. Represent the unknown value using a variable, such as x.
  3. Translate the phrase into a mathematical expression using the variable and the identified elements.
  4. Use the correct inequality symbol to represent the comparison.

Q: What are the different types of inequality symbols?

A: There are four main types of inequality symbols:

  • ≥ (greater than or equal to)
  • ≤ (less than or equal to)

  • (greater than)

  • < (less than)

Each symbol represents a different type of comparison, and it's essential to use the correct symbol when writing an inequality.

Q: How do I solve an inequality?

A: To solve an inequality, follow these steps:

  1. Isolate the variable on one side of the inequality.
  2. Use algebraic manipulation to simplify the inequality.
  3. Use the correct inequality symbol to represent the solution.

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

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

  • Using the wrong inequality symbol.
  • Failing to isolate the variable.
  • Not considering the direction of the inequality.
  • Not checking the solution for validity.

Q: How do I check the solution for an inequality?

A: To check the solution for an inequality, substitute the solution into the original inequality and verify that it's true. If the solution satisfies the inequality, it's a valid solution.

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

A: Inequalities have numerous real-world applications, including:

  • Modeling population growth and decline.
  • Representing financial transactions and budgets.
  • Solving optimization problems in business and economics.
  • Analyzing data and making predictions in science and engineering.

Conclusion

Writing inequalities from phrases is a fundamental skill in mathematics, and it's essential to understand the concepts and techniques involved. By following the steps outlined in this article and avoiding common mistakes, you can master the art of writing and solving inequalities. Remember to practice regularly and apply your skills to real-world problems to become proficient in this area.

Key Takeaways

  • Inequalities are mathematical expressions that compare two values.
  • Writing inequalities from phrases involves identifying key elements and using the correct inequality symbol.
  • Solving inequalities involves isolating the variable and using algebraic manipulation.
  • Checking the solution for an inequality involves substituting the solution into the original inequality.

Additional Resources

For further practice and review, try the following resources:

  • Online inequality worksheets and exercises.
  • Video tutorials and lectures on writing and solving inequalities.
  • Practice problems and quizzes to test your skills.
  • Real-world applications and case studies to apply your knowledge.