Solve The Compound Inequality:${ 2x - 7 \ \textless \ 3 \ \textless \ 27 + 4x }$

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Introduction

Compound inequalities are a type of mathematical expression that involves multiple inequalities joined by logical operators, such as "and" or "or." In this article, we will focus on solving compound inequalities of the form a<b<ca < b < c, where aa, bb, and cc are algebraic expressions. We will use the given compound inequality 2x−7<3<27+4x2x - 7 < 3 < 27 + 4x as an example to illustrate the steps involved in solving such inequalities.

Understanding Compound Inequalities

A compound inequality is a statement that combines two or more inequalities using logical operators. In the given compound inequality, we have two inequalities: 2x−7<32x - 7 < 3 and 3<27+4x3 < 27 + 4x. The logical operator "and" is implied between these two inequalities, meaning that both inequalities must be true simultaneously.

Solving the First Inequality

The first inequality is 2x−7<32x - 7 < 3. To solve this inequality, we need to isolate the variable xx on one side of the inequality sign. We can do this by adding 7 to both sides of the inequality:

2x−7+7<3+72x - 7 + 7 < 3 + 7

This simplifies to:

2x<102x < 10

Next, we can divide both sides of the inequality by 2 to solve for xx:

2x2<102\frac{2x}{2} < \frac{10}{2}

This gives us:

x<5x < 5

Solving the Second Inequality

The second inequality is 3<27+4x3 < 27 + 4x. To solve this inequality, we need to isolate the variable xx on one side of the inequality sign. We can do this by subtracting 27 from both sides of the inequality:

3−27<4x−273 - 27 < 4x - 27

This simplifies to:

−24<4x−27-24 < 4x - 27

Next, we can add 27 to both sides of the inequality to get:

−24+27<4x−27+27-24 + 27 < 4x - 27 + 27

This gives us:

3<4x3 < 4x

Combining the Two Inequalities

Now that we have solved both inequalities, we can combine them to get the final solution. We need to find the values of xx that satisfy both inequalities simultaneously. From the first inequality, we have x<5x < 5, and from the second inequality, we have 3<4x3 < 4x.

Finding the Intersection of the Two Inequalities

To find the intersection of the two inequalities, we can rewrite the second inequality as x>34x > \frac{3}{4}. This gives us the following compound inequality:

34<x<5\frac{3}{4} < x < 5

This means that the values of xx that satisfy both inequalities simultaneously are those that lie between 34\frac{3}{4} and 5.

Conclusion

Solving compound inequalities requires a step-by-step approach. We need to isolate the variable on one side of the inequality sign and then combine the two inequalities to find the final solution. In this article, we used the compound inequality 2x−7<3<27+4x2x - 7 < 3 < 27 + 4x as an example to illustrate the steps involved in solving such inequalities. We found that the values of xx that satisfy both inequalities simultaneously are those that lie between 34\frac{3}{4} and 5.

Example Problems

Problem 1

Solve the compound inequality x−2<3<2x+5x - 2 < 3 < 2x + 5.

Solution

To solve this inequality, we need to isolate the variable xx on one side of the inequality sign. We can do this by adding 2 to both sides of the first inequality and subtracting 5 from both sides of the second inequality:

x−2+2<3+2x - 2 + 2 < 3 + 2

This simplifies to:

x<5x < 5

3−5<2x+5−53 - 5 < 2x + 5 - 5

This gives us:

−2<2x-2 < 2x

Next, we can divide both sides of the inequality by 2 to solve for xx:

−22<2x2\frac{-2}{2} < \frac{2x}{2}

This gives us:

−1<x-1 < x

Now that we have solved both inequalities, we can combine them to get the final solution. We need to find the values of xx that satisfy both inequalities simultaneously. From the first inequality, we have x<5x < 5, and from the second inequality, we have −1<x-1 < x.

Finding the Intersection of the Two Inequalities

To find the intersection of the two inequalities, we can rewrite the second inequality as x>−1x > -1. This gives us the following compound inequality:

−1<x<5-1 < x < 5

This means that the values of xx that satisfy both inequalities simultaneously are those that lie between -1 and 5.

Problem 2

Solve the compound inequality 2x+3<5<x+72x + 3 < 5 < x + 7.

Solution

To solve this inequality, we need to isolate the variable xx on one side of the inequality sign. We can do this by subtracting 3 from both sides of the first inequality and subtracting 7 from both sides of the second inequality:

2x+3−3<5−32x + 3 - 3 < 5 - 3

This simplifies to:

2x<22x < 2

5−7<x+7−75 - 7 < x + 7 - 7

This gives us:

−2<x-2 < x

Next, we can divide both sides of the inequality by 2 to solve for xx:

2x2<22\frac{2x}{2} < \frac{2}{2}

This gives us:

x<1x < 1

Now that we have solved both inequalities, we can combine them to get the final solution. We need to find the values of xx that satisfy both inequalities simultaneously. From the first inequality, we have x<1x < 1, and from the second inequality, we have −2<x-2 < x.

Finding the Intersection of the Two Inequalities

To find the intersection of the two inequalities, we can rewrite the second inequality as x>−2x > -2. This gives us the following compound inequality:

−2<x<1-2 < x < 1

This means that the values of xx that satisfy both inequalities simultaneously are those that lie between -2 and 1.

Final Thoughts

Solving compound inequalities requires a step-by-step approach. We need to isolate the variable on one side of the inequality sign and then combine the two inequalities to find the final solution. In this article, we used the compound inequality 2x−7<3<27+4x2x - 7 < 3 < 27 + 4x as an example to illustrate the steps involved in solving such inequalities. We found that the values of xx that satisfy both inequalities simultaneously are those that lie between 34\frac{3}{4} and 5. We also provided two example problems to demonstrate the application of the steps involved in solving compound inequalities.

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Q: What is a compound inequality?

A: A compound inequality is a type of mathematical expression that involves multiple inequalities joined by logical operators, such as "and" or "or." In this article, we will focus on solving compound inequalities of the form a<b<ca < b < c, where aa, bb, and cc are algebraic expressions.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to isolate the variable on one side of the inequality sign and then combine the two inequalities to find the final solution. This involves solving each inequality separately and then finding the intersection of the two inequalities.

Q: What is the intersection of two inequalities?

A: The intersection of two inequalities is the set of values that satisfy both inequalities simultaneously. To find the intersection, you need to rewrite the second inequality as x>ax > a or x<bx < b, where aa and bb are the values obtained from the first inequality.

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

A: To find the intersection of two inequalities, you need to rewrite the second inequality as x>ax > a or x<bx < b, where aa and bb are the values obtained from the first inequality. Then, you can combine the two inequalities to get the final solution.

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

A: Some common mistakes to avoid when solving compound inequalities include:

  • Not isolating the variable on one side of the inequality sign
  • Not combining the two inequalities to find the final solution
  • Not rewriting the second inequality as x>ax > a or x<bx < b
  • Not checking the solution for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solution back into the original inequality to see if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

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

A: Compound inequalities have many real-world applications, including:

  • Modeling population growth and decline
  • Analyzing financial data
  • Solving optimization problems
  • Modeling physical systems

Q: How do I apply compound inequalities to real-world problems?

A: To apply compound inequalities to real-world problems, you need to:

  • Identify the variables and constants in the problem
  • Write the compound inequality based on the problem
  • Solve the compound inequality to find the solution
  • Check the solution for extraneous solutions

Q: What are some tips for solving compound inequalities?

A: Some tips for solving compound inequalities include:

  • Start by isolating the variable on one side of the inequality sign
  • Use algebraic properties to simplify the inequality
  • Check the solution for extraneous solutions
  • Use a graphing calculator to visualize the solution

Q: How do I use a graphing calculator to visualize the solution?

A: To use a graphing calculator to visualize the solution, you need to:

  • Enter the compound inequality into the calculator
  • Graph the inequality to see the solution
  • Use the calculator to check for extraneous solutions

Q: What are some common mistakes to avoid when using a graphing calculator?

A: Some common mistakes to avoid when using a graphing calculator include:

  • Not entering the compound inequality correctly
  • Not graphing the inequality correctly
  • Not checking for extraneous solutions

Q: How do I choose the right graphing calculator for my needs?

A: To choose the right graphing calculator for your needs, you need to:

  • Consider the features and functions of the calculator
  • Read reviews and compare different calculators
  • Ask for recommendations from teachers or peers

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

A: Some additional resources for learning about compound inequalities include:

  • Online tutorials and videos
  • Textbooks and workbooks
  • Online communities and forums
  • Graphing calculator software and apps

Q: How do I get help if I'm struggling with compound inequalities?

A: If you're struggling with compound inequalities, you can:

  • Ask your teacher or tutor for help
  • Seek help from a classmate or peer
  • Use online resources and tutorials
  • Practice problems and exercises to build your skills and confidence.