Solve The Compound Inequality. 2 X − 2 \textless 4 And 3 X − 5 ≥ 13 2x - 2 \ \textless \ 4 \quad \text{and} \quad 3x - 5 \geq 13 2 X − 2 \textless 4 And 3 X − 5 ≥ 13 Write The Solution In Interval Notation. If There Is No Solution, Enter ∅ \varnothing ∅ .

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Introduction

Compound inequalities are a combination of two or more inequalities that are connected by logical operators such as "and" or "or". In this article, we will focus on solving compound inequalities of the form a≤x≤ba \leq x \leq b and c≥x≥dc \geq x \geq d, where aa, bb, cc, and dd are constants. We will use the given compound inequality 2x−2<42x - 2 < 4 and 3x−5≥133x - 5 \geq 13 as an example to demonstrate the steps involved in solving compound inequalities.

Step 1: Solve the First Inequality

The first inequality is 2x−2<42x - 2 < 4. To solve this inequality, we need to isolate the variable xx.

Adding 2 to Both Sides

We add 2 to both sides of the inequality to get:

2x−2+2<4+22x - 2 + 2 < 4 + 2

This simplifies to:

2x<62x < 6

Dividing Both Sides by 2

We divide both sides of the inequality by 2 to get:

2x2<62\frac{2x}{2} < \frac{6}{2}

This simplifies to:

x<3x < 3

Step 2: Solve the Second Inequality

The second inequality is 3x−5≥133x - 5 \geq 13. To solve this inequality, we need to isolate the variable xx.

Adding 5 to Both Sides

We add 5 to both sides of the inequality to get:

3x−5+5≥13+53x - 5 + 5 \geq 13 + 5

This simplifies to:

3x≥183x \geq 18

Dividing Both Sides by 3

We divide both sides of the inequality by 3 to get:

3x3≥183\frac{3x}{3} \geq \frac{18}{3}

This simplifies to:

x≥6x \geq 6

Step 3: Find the Intersection of the Two Solutions

Now that we have solved both inequalities, we need to find the intersection of the two solutions. The first inequality has a solution of x<3x < 3, and the second inequality has a solution of x≥6x \geq 6. To find the intersection, we need to find the values of xx that satisfy both inequalities.

Intersection of x<3x < 3 and x≥6x \geq 6

Since the two inequalities have no common values, the intersection is empty. Therefore, the solution to the compound inequality is ∅\varnothing.

Conclusion

In this article, we have demonstrated the steps involved in solving compound inequalities. We used the given compound inequality 2x−2<42x - 2 < 4 and 3x−5≥133x - 5 \geq 13 as an example to show how to solve each inequality separately and then find the intersection of the two solutions. The solution to the compound inequality is ∅\varnothing, indicating that there is no value of xx that satisfies both inequalities.

Frequently Asked Questions

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities that are connected by logical operators such as "and" or "or".

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately and then find the intersection of the two solutions.

Q: What is the intersection of two inequalities?

A: The intersection of two inequalities is the set of values that satisfy both inequalities.

Q: How do I find the intersection of two inequalities?

A: To find the intersection of two inequalities, you need to find the values of the variable that satisfy both inequalities.

Example Problems

Problem 1

Solve the compound inequality x−2<3x - 2 < 3 and 2x+1≥72x + 1 \geq 7.

Solution

To solve the compound inequality, we need to solve each inequality separately.

  • The first inequality is x−2<3x - 2 < 3. To solve this inequality, we add 2 to both sides to get x<5x < 5.
  • The second inequality is 2x+1≥72x + 1 \geq 7. To solve this inequality, we subtract 1 from both sides to get 2x≥62x \geq 6, and then divide both sides by 2 to get x≥3x \geq 3.
  • The intersection of the two solutions is the set of values that satisfy both inequalities. Since the two inequalities have no common values, the intersection is empty. Therefore, the solution to the compound inequality is ∅\varnothing.

Problem 2

Solve the compound inequality x+1≤4x + 1 \leq 4 and x−2≥1x - 2 \geq 1.

Solution

To solve the compound inequality, we need to solve each inequality separately.

  • The first inequality is x+1≤4x + 1 \leq 4. To solve this inequality, we subtract 1 from both sides to get x≤3x \leq 3.
  • The second inequality is x−2≥1x - 2 \geq 1. To solve this inequality, we add 2 to both sides to get x≥3x \geq 3.
  • The intersection of the two solutions is the set of values that satisfy both inequalities. Since the two inequalities have common values, the intersection is the set of values that satisfy both inequalities. Therefore, the solution to the compound inequality is x≥3x \geq 3.

Final Answer

The final answer is ∅\varnothing.

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Q: What is a compound inequality?

A compound inequality is a combination of two or more inequalities that are connected by logical operators such as "and" or "or".

Q: How do I solve a compound inequality?

To solve a compound inequality, you need to solve each inequality separately and then find the intersection of the two solutions.

Q: What is the intersection of two inequalities?

The intersection of two inequalities is the set of values that satisfy both inequalities.

Q: How do I find the intersection of two inequalities?

To find the intersection of two inequalities, you need to find the values of the variable that satisfy both inequalities.

Q: What is the difference between a compound inequality and a single inequality?

A single inequality is a statement that compares a single value to a variable, such as x>3x > 3. A compound inequality is a combination of two or more inequalities that are connected by logical operators such as "and" or "or".

Q: Can a compound inequality have more than two inequalities?

Yes, a compound inequality can have more than two inequalities. For example, the compound inequality x−2<3x - 2 < 3 and 2x+1≥72x + 1 \geq 7 and x+2≤9x + 2 \leq 9 is a combination of three inequalities.

Q: How do I solve a compound inequality with more than two inequalities?

To solve a compound inequality with more than two inequalities, you need to solve each inequality separately and then find the intersection of the two solutions.

Q: What is the importance of solving compound inequalities?

Solving compound inequalities is important because it helps you to find the values of the variable that satisfy multiple conditions. This is useful in many real-world applications, such as solving systems of equations and inequalities.

Q: Can a compound inequality have no solution?

Yes, a compound inequality can have no solution. This occurs when the intersection of the two solutions is empty.

Q: How do I determine if a compound inequality has no solution?

To determine if a compound inequality has no solution, you need to find the intersection of the two solutions. If the intersection is empty, then the compound inequality has no solution.

Q: What is the difference between a compound inequality and a system of inequalities?

A system of inequalities is a set of two or more inequalities that are connected by logical operators such as "and" or "or". A compound inequality is a single inequality that is a combination of two or more inequalities.

Q: Can a compound inequality be used to solve a system of inequalities?

Yes, a compound inequality can be used to solve a system of inequalities. By solving the compound inequality, you can find the values of the variable that satisfy multiple conditions.

Q: How do I use a compound inequality to solve a system of inequalities?

To use a compound inequality to solve a system of inequalities, you need to combine the inequalities into a single compound inequality and then solve the compound inequality.

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

Some real-world applications of compound inequalities include:

  • Solving systems of equations and inequalities
  • Finding the values of a variable that satisfy multiple conditions
  • Modeling real-world problems that involve multiple conditions
  • Solving optimization problems that involve multiple constraints

Q: Can compound inequalities be used in other areas of mathematics?

Yes, compound inequalities can be used in other areas of mathematics, such as:

  • Algebra
  • Geometry
  • Calculus
  • Statistics

Q: How do I learn more about compound inequalities?

To learn more about compound inequalities, you can:

  • Read textbooks and online resources
  • Watch video tutorials and online lectures
  • Practice solving compound inequalities with examples and exercises
  • Join online communities and forums to discuss compound inequalities with other mathematicians

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

Some common mistakes to avoid when solving compound inequalities include:

  • Failing to solve each inequality separately
  • Failing to find the intersection of the two solutions
  • Making errors when combining the inequalities into a single compound inequality
  • Failing to check for extraneous solutions

Q: How do I avoid common mistakes when solving compound inequalities?

To avoid common mistakes when solving compound inequalities, you need to:

  • Carefully read and understand the problem
  • Solve each inequality separately
  • Find the intersection of the two solutions
  • Check for extraneous solutions
  • Double-check your work for errors

Q: What are some tips for solving compound inequalities?

Some tips for solving compound inequalities include:

  • Start by solving each inequality separately
  • Use a systematic approach to find the intersection of the two solutions
  • Check for extraneous solutions
  • Double-check your work for errors
  • Practice solving compound inequalities with examples and exercises

Q: How do I practice solving compound inequalities?

To practice solving compound inequalities, you can:

  • Use online resources and practice problems
  • Work with a study group or tutor
  • Join online communities and forums to discuss compound inequalities with other mathematicians
  • Practice solving compound inequalities with examples and exercises

Q: What are some resources for learning about compound inequalities?

Some resources for learning about compound inequalities include:

  • Textbooks and online resources
  • Video tutorials and online lectures
  • Online communities and forums
  • Practice problems and exercises
  • Study guides and review materials