Solve The Inequality:${2(4x - 3) \geq -3(3x) + 5x}$A. { X \geq 0.5$}$ B. { X \geq 2$}$ C. { (- \infty, 0.5]$}$ D. { (- \infty, 2]$}$

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Introduction

Inequalities are mathematical expressions that compare two values, often using greater than or less than symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the relationship between the two sides. In this article, we will focus on solving the inequality 2(4x−3)≥−3(3x)+5x2(4x - 3) \geq -3(3x) + 5x and explore the different methods and techniques used to solve inequalities.

Understanding the Inequality

The given inequality is 2(4x−3)≥−3(3x)+5x2(4x - 3) \geq -3(3x) + 5x. To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expression on both sides.

Step 1: Distribute the Numbers

First, we need to distribute the numbers inside the parentheses.

2(4x−3)=8x−62(4x - 3) = 8x - 6

−3(3x)=−9x-3(3x) = -9x

5x=5x5x = 5x

So, the inequality becomes:

8x−6≥−9x+5x8x - 6 \geq -9x + 5x

Step 2: Combine Like Terms

Next, we need to combine like terms on both sides of the inequality.

8x−6≥−4x+5x8x - 6 \geq -4x + 5x

8x−6≥x8x - 6 \geq x

Step 3: Isolate the Variable

Now, we need to isolate the variable x on one side of the inequality.

8x−x≥68x - x \geq 6

7x≥67x \geq 6

Step 4: Divide Both Sides

Finally, we need to divide both sides of the inequality by 7 to solve for x.

x≥67x \geq \frac{6}{7}

Analyzing the Solution

Now that we have solved the inequality, let's analyze the solution.

x≥67x \geq \frac{6}{7}

This means that x is greater than or equal to 6/7. In other words, x is greater than or equal to 0.8571 (rounded to four decimal places).

Comparing with the Options

Let's compare our solution with the given options.

A. x≥0.5x \geq 0.5

B. x≥2x \geq 2

C. (−∞,0.5](- \infty, 0.5]

D. (−∞,2](- \infty, 2]

Our solution, x≥67x \geq \frac{6}{7}, is closest to option A, x≥0.5x \geq 0.5. However, our solution is not exactly equal to option A, as it is a more specific and accurate solution.

Conclusion

Solving inequalities involves following a series of steps, including distributing numbers, combining like terms, isolating the variable, and dividing both sides. By following these steps, we can solve complex inequalities and find the solution set. In this article, we solved the inequality 2(4x−3)≥−3(3x)+5x2(4x - 3) \geq -3(3x) + 5x and found that the solution is x≥67x \geq \frac{6}{7}. This solution is closest to option A, x≥0.5x \geq 0.5, but is a more specific and accurate solution.

Frequently Asked Questions

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a mnemonic device that helps us remember the order of operations in mathematics. PEMDAS stands for:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division (from left to right)
  4. Addition and Subtraction (from left to right)

Q: How do I solve inequalities?

A: To solve inequalities, follow these steps:

  1. Distribute numbers inside parentheses
  2. Combine like terms
  3. Isolate the variable
  4. Divide both sides (if necessary)

Q: What is the difference between an inequality and an equation?

A: An inequality is a mathematical expression that compares two values using greater than or less than symbols. An equation is a mathematical expression that states that two values are equal.

Q: How do I graph an inequality?

A: To graph an inequality, follow these steps:

  1. Draw a number line
  2. Plot a point on the number line that satisfies the inequality
  3. Shade the region on the number line that satisfies the inequality

Final Answer

The final answer is: A\boxed{A}

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Introduction

In our previous article, we solved the inequality 2(4x−3)≥−3(3x)+5x2(4x - 3) \geq -3(3x) + 5x and found that the solution is x≥67x \geq \frac{6}{7}. In this article, we will answer some frequently asked questions about solving inequalities.

Q&A

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a mnemonic device that helps us remember the order of operations in mathematics. PEMDAS stands for:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division (from left to right)
  4. Addition and Subtraction (from left to right)

Q: How do I solve inequalities?

A: To solve inequalities, follow these steps:

  1. Distribute numbers inside parentheses
  2. Combine like terms
  3. Isolate the variable
  4. Divide both sides (if necessary)

Q: What is the difference between an inequality and an equation?

A: An inequality is a mathematical expression that compares two values using greater than or less than symbols. An equation is a mathematical expression that states that two values are equal.

Q: How do I graph an inequality?

A: To graph an inequality, follow these steps:

  1. Draw a number line
  2. Plot a point on the number line that satisfies the inequality
  3. Shade the region on the number line that satisfies the inequality

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values that satisfy the inequality. For example, if we have the inequality x≥2x \geq 2, the solution set is all values greater than or equal to 2.

Q: How do I determine the direction of the inequality?

A: To determine the direction of the inequality, follow these steps:

  1. Check if the inequality is greater than or equal to (≥) or less than or equal to (≤)
  2. If the inequality is greater than or equal to (≥), the solution set will be all values greater than or equal to the value on the right-hand side
  3. If the inequality is less than or equal to (≤), the solution set will be all values less than or equal to the value on the right-hand side

Q: Can I have multiple solution sets for an inequality?

A: Yes, it is possible to have multiple solution sets for an inequality. For example, if we have the inequality x≥2x \geq 2 and x≤−3x \leq -3, the solution set will be all values greater than or equal to 2 and less than or equal to -3.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, follow these steps:

  1. Solve each inequality separately
  2. Find the intersection of the solution sets
  3. The solution set of the system of inequalities will be the intersection of the solution sets of each inequality

Tips and Tricks

Tip 1: Always check your work

When solving inequalities, it's essential to check your work to ensure that you have the correct solution set.

Tip 2: Use a number line to visualize the solution set

Using a number line can help you visualize the solution set and make it easier to understand.

Tip 3: Practice, practice, practice

Solving inequalities takes practice, so make sure to practice regularly to become more comfortable with the process.

Conclusion

Solving inequalities can be a challenging task, but with practice and patience, you can become more confident and proficient. Remember to always check your work, use a number line to visualize the solution set, and practice regularly. By following these tips and tricks, you'll be well on your way to becoming an expert in solving inequalities.

Final Answer

The final answer is: A\boxed{A}