Solve The Inequality: $ 2(4 + 2x) \geq 5x + 5 }$Options 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 is a crucial skill in mathematics, as it helps us understand and analyze various real-world problems. In this article, we will focus on solving the inequality 2(4+2x)≥5x+52(4 + 2x) \geq 5x + 5 and explore the different options provided.

Understanding the Inequality

The given inequality is 2(4+2x)≥5x+52(4 + 2x) \geq 5x + 5. To solve this inequality, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: None in this case.
  3. Multiplication and Division: Multiply and divide from left to right.
  4. Addition and Subtraction: Add and subtract from left to right.

Step 1: Distribute the 2

Distribute the 2 to the terms inside the parentheses:

2(4+2x)=8+4x2(4 + 2x) = 8 + 4x

So, the inequality becomes:

8+4x≥5x+58 + 4x \geq 5x + 5

Step 2: Subtract 5x from Both Sides

Subtract 5x from both sides of the inequality to isolate the x term:

8+4x−5x≥5x−5x+58 + 4x - 5x \geq 5x - 5x + 5

This simplifies to:

8−x≥58 - x \geq 5

Step 3: Add x to Both Sides

Add x to both sides of the inequality to further isolate the x term:

8−x+x≥5+x8 - x + x \geq 5 + x

This simplifies to:

8≥5+x8 \geq 5 + x

Step 4: Subtract 5 from Both Sides

Subtract 5 from both sides of the inequality to isolate the x term:

8−5≥5+x−58 - 5 \geq 5 + x - 5

This simplifies to:

3≥x3 \geq x

Step 5: Write the Solution in Interval Notation

The solution to the inequality is x≤3x \leq 3. This can be written in interval notation as (−∞,3](-\infty, 3].

Conclusion

In conclusion, the solution to the inequality 2(4+2x)≥5x+52(4 + 2x) \geq 5x + 5 is x≤3x \leq 3. This means that any value of x less than or equal to 3 satisfies the inequality.

Comparison with Options

Let's compare our solution with the options provided:

  • Option A: x≤−2x \leq -2
  • Option B: x≥−2x \geq -2
  • Option C: x≤3x \leq 3
  • Option D: x≥3x \geq 3

Our solution, x≤3x \leq 3, matches option C.

Final Answer

The final answer is:

  • Option C: x≤3x \leq 3

This is the correct solution to the inequality 2(4+2x)≥5x+52(4 + 2x) \geq 5x + 5.

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Q&A: Frequently Asked Questions about 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: How do I solve an inequality?

A: To solve an inequality, follow the order of operations (PEMDAS) and isolate the variable (x) on one side of the inequality.

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

A: A linear inequality is an inequality that can be written in the form ax + b ≥ c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c ≥ d, where a, b, c, and d are constants.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, draw a line to represent the boundary of the inequality. If the inequality is greater than or equal to, use a closed circle to represent the boundary. If the inequality is less than or equal to, use an open circle to represent the boundary.

Q: Can I use the same steps to solve a system of inequalities as I would to solve a system of equations?

A: No, the steps to solve a system of inequalities are different from the steps to solve a system of equations. When solving a system of inequalities, you need to find the intersection of the solution sets of each inequality.

Q: How do I determine the solution set of a system of inequalities?

A: To determine the solution set of a system of inequalities, graph each inequality on a number line and find the intersection of the solution sets.

Q: Can I use algebraic methods to solve a system of inequalities?

A: Yes, you can use algebraic methods to solve a system of inequalities. One common method is to use the method of substitution or elimination to solve the system.

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

A: A linear programming problem is a problem that involves maximizing or minimizing a linear objective function subject to a set of linear constraints. A quadratic programming problem is a problem that involves maximizing or minimizing a quadratic objective function subject to a set of linear constraints.

Q: How do I solve a linear programming problem?

A: To solve a linear programming problem, use the method of linear programming, which involves finding the optimal solution to the problem by using linear programming techniques such as the simplex method.

Q: Can I use a calculator to solve a system of inequalities?

A: Yes, you can use a calculator to solve a system of inequalities. Many calculators have built-in functions for solving systems of inequalities.

Q: How do I determine the solution set of a system of inequalities using a calculator?

A: To determine the solution set of a system of inequalities using a calculator, enter the inequalities into the calculator and use the built-in functions to solve the system.

Conclusion

In conclusion, solving inequalities is a crucial skill in mathematics, and understanding the different types of inequalities and how to solve them is essential for success in mathematics and other fields. By following the steps outlined in this article and using the Q&A section to answer frequently asked questions, you can become proficient in solving inequalities and apply this skill to real-world problems.

Final Tips

  • Practice solving inequalities regularly to become proficient in this skill.
  • Use a calculator to check your work and ensure that you are getting the correct solution.
  • Apply the skills you learn in this article to real-world problems to see the relevance and importance of solving inequalities.

Resources

  • Khan Academy: Inequalities
  • Mathway: Inequalities
  • Wolfram Alpha: Inequalities

Final Answer

The final answer is:

  • Solving inequalities is a crucial skill in mathematics that involves comparing two values using greater than, less than, or equal to symbols.