Solve The Inequality: $ 2(4+2x) \geq 5x+5 $A. $ X \leq -2 $ B. $ X \geq -2 $ C. $ X \leq 3 $ D. $ X \geq 3 $

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Introduction

Inequalities are mathematical expressions that compare two values, often using greater than, less than, or equal to symbols. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality 2(4+2x)≥5x+52(4+2x) \geq 5x+5. We will break down the solution into manageable steps and provide a clear explanation of each step.

Step 1: Distribute the Coefficient

The first step in solving the inequality is to distribute the coefficient 2 to the terms inside the parentheses.

2(4+2x)≥5x+52(4+2x) \geq 5x+5

Using the distributive property, we get:

8+4x≥5x+58+4x \geq 5x+5

Step 2: Subtract 5x from Both Sides

Next, we want to isolate the variable x on one side of the inequality. To do this, we subtract 5x from both sides of the inequality.

8+4x−5x≥5x+5−5x8+4x-5x \geq 5x+5-5x

Simplifying the left-hand side, we get:

8−x≥58-x \geq 5

Step 3: Add x to Both Sides

Now, we want to isolate the constant term on the right-hand side of the inequality. To do this, we add x to both sides of the inequality.

8−x+x≥5+x8-x+x \geq 5+x

Simplifying the left-hand side, we get:

8≥5+x8 \geq 5+x

Step 4: Subtract 5 from Both Sides

Next, we want to isolate the constant term on the right-hand side of the inequality. To do this, we subtract 5 from both sides of the inequality.

8−5≥5+x−58-5 \geq 5+x-5

Simplifying the left-hand side, we get:

3≥x3 \geq x

Step 5: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. Since x is less than or equal to 3, we can write the solution as:

x≤3x \leq 3

Conclusion

Solving inequalities involves breaking down the solution into manageable steps and using algebraic properties to isolate the variable. In this article, we solved the inequality 2(4+2x)≥5x+52(4+2x) \geq 5x+5 and found that the solution is x≤3x \leq 3. We hope this article has provided a clear and concise explanation of how to solve inequalities.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes to avoid. These include:

  • Not distributing the coefficient: Failing to distribute the coefficient to the terms inside the parentheses can lead to incorrect solutions.
  • Not isolating the variable: Failing to isolate the variable on one side of the inequality can lead to incorrect solutions.
  • Not checking the direction of the inequality: Failing to check the direction of the inequality can lead to incorrect solutions.

Tips and Tricks

When solving inequalities, there are several tips and tricks to keep in mind. These include:

  • Use algebraic properties: Algebraic properties such as the distributive property and the commutative property can help simplify the inequality and make it easier to solve.
  • Check the direction of the inequality: Checking the direction of the inequality can help ensure that the solution is correct.
  • Use interval notation: Using interval notation can help write the solution in a clear and concise manner.

Real-World Applications

Solving inequalities has several real-world applications. These include:

  • Finance: In finance, inequalities are used to model financial situations such as investments and loans.
  • Science: In science, inequalities are used to model physical situations such as motion and energy.
  • Engineering: In engineering, inequalities are used to model complex systems and ensure that they are safe and efficient.

Conclusion

Introduction

In our previous article, we discussed how to solve inequalities using algebraic properties and interval notation. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often using greater than, less than, or equal to symbols.

Q: What are the different types of inequalities?

A: There are two main types of inequalities: linear inequalities and quadratic inequalities. Linear inequalities involve a single variable and a linear expression, while quadratic inequalities involve a single variable and a quadratic expression.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can use the following steps:

  1. Distribute the coefficient to the terms inside the parentheses.
  2. Add or subtract the same value to both sides of the inequality.
  3. Multiply or divide both sides of the inequality by the same non-zero value.
  4. Write the solution in interval notation.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

  1. Factor the quadratic expression, if possible.
  2. Use the factored form to identify the critical points.
  3. Test each interval to determine which intervals satisfy the inequality.
  4. Write the solution in interval notation.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality involves a single variable and a linear expression, while a quadratic inequality involves a single variable and a quadratic expression. Quadratic inequalities are more complex and require additional techniques, such as factoring and testing intervals.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you can use the following steps:

  1. Plug in a value from each interval to test whether it satisfies the inequality.
  2. Use a graphing calculator or software to visualize the solution.
  3. Check your work by plugging in a value from each interval and verifying that it satisfies the inequality.

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

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

  • Not distributing the coefficient to the terms inside the parentheses.
  • Not isolating the variable on one side of the inequality.
  • Not checking the direction of the inequality.
  • Not using interval notation to write the solution.

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

A: Inequalities have many real-world applications, including finance, science, and engineering. To apply inequalities to real-world problems, you can use the following steps:

  1. Identify the variables and constants in the problem.
  2. Write an inequality that represents the problem.
  3. Solve the inequality using algebraic properties and interval notation.
  4. Interpret the solution in the context of the problem.

Conclusion

Solving inequalities is an important skill that has many real-world applications. By following the steps outlined in this article and avoiding mistakes, you can become proficient in solving inequalities and apply them to real-world problems. Remember to check your work and use interval notation to write the solution. With practice and patience, you can become a master of solving inequalities.