Solve The Inequality:${ 8\left(\frac{1}{2} X - \frac{1}{4}\right) \ \textgreater \ 12 - 2x }$A. ${ X \ \textgreater \ \frac{5}{3} }$ B. ${ X \ \textgreater \ 7 }$ C. $[ X \ \textgreater \ \frac{7}{3}

by ADMIN 213 views

Introduction

Inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will focus on solving the given inequality: 8(12x−14) \textgreater 12−2x8\left(\frac{1}{2} x - \frac{1}{4}\right) \ \textgreater \ 12 - 2x. 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 88 to the terms inside the parentheses. This will allow us to simplify the expression and make it easier to work with.

8(12x−14) \textgreater 12−2x{ 8\left(\frac{1}{2} x - \frac{1}{4}\right) \ \textgreater \ 12 - 2x }

4x−2 \textgreater 12−2x{ 4x - 2 \ \textgreater \ 12 - 2x }

Step 2: Add 2x to Both Sides

The next step is to add 2x2x to both sides of the inequality. This will help us to isolate the variable xx on one side of the inequality.

4x−2+2x \textgreater 12−2x+2x{ 4x - 2 + 2x \ \textgreater \ 12 - 2x + 2x }

6x−2 \textgreater 12{ 6x - 2 \ \textgreater \ 12 }

Step 3: Add 2 to Both Sides

The next step is to add 22 to both sides of the inequality. This will help us to eliminate the constant term on the left-hand side of the inequality.

6x−2+2 \textgreater 12+2{ 6x - 2 + 2 \ \textgreater \ 12 + 2 }

6x \textgreater 14{ 6x \ \textgreater \ 14 }

Step 4: Divide Both Sides by 6

The final step is to divide both sides of the inequality by 66. This will help us to isolate the variable xx on one side of the inequality.

6x6 \textgreater 146{ \frac{6x}{6} \ \textgreater \ \frac{14}{6} }

x \textgreater 73{ x \ \textgreater \ \frac{7}{3} }

Conclusion

In conclusion, the solution to the inequality 8(12x−14) \textgreater 12−2x8\left(\frac{1}{2} x - \frac{1}{4}\right) \ \textgreater \ 12 - 2x is x \textgreater 73x \ \textgreater \ \frac{7}{3}. This means that the value of xx must be greater than 73\frac{7}{3} in order to satisfy the inequality.

Comparison with Other Options

Let's compare our solution with the other options provided:

  • Option A: x \textgreater 53x \ \textgreater \ \frac{5}{3}
  • Option B: x \textgreater 7x \ \textgreater \ 7
  • Option C: x \textgreater 73x \ \textgreater \ \frac{7}{3}

Our solution, x \textgreater 73x \ \textgreater \ \frac{7}{3}, is the only option that matches the correct solution. Therefore, the correct answer is:

The Correct Answer is Option C: x \textgreater 73x \ \textgreater \ \frac{7}{3}

Final Thoughts

Introduction

In our previous article, we solved the inequality 8(12x−14) \textgreater 12−2x8\left(\frac{1}{2} x - \frac{1}{4}\right) \ \textgreater \ 12 - 2x and arrived at the correct solution: x \textgreater 73x \ \textgreater \ \frac{7}{3}. In this article, we will provide a Q&A guide to help you understand the concept of solving inequalities better.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, indicating whether one value is greater than, less than, or equal to another value.

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 need to isolate the variable on one side of the inequality sign, while maintaining the direction of the inequality. This can be done by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

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 the use of the quadratic formula to solve.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression, if possible, and then use the quadratic formula to find the solutions. You can also use the graphing method to visualize the solutions.

Q: What is the graphing method?

A: The graphing method involves graphing the quadratic expression on a coordinate plane and then identifying the intervals where the inequality is true.

Q: What are the different types of solutions to an inequality?

A: The different types of solutions to an inequality are:

  • Open interval: An open interval is a set of values that are greater than or less than a certain value.
  • Closed interval: A closed interval is a set of values that are equal to or greater than or less than a certain value.
  • Half-open interval: A half-open interval is a set of values that are greater than or less than a certain value, but not equal to it.

Q: How do I determine the type of solution to an inequality?

A: To determine the type of solution to an inequality, you need to look at the inequality sign and the values that are being compared. If the inequality sign is greater than or less than, the solution is an open interval. If the inequality sign is equal to or greater than or less than, the solution is a closed interval. If the inequality sign is greater than or less than, but not equal to, the solution is a half-open interval.

Conclusion

In conclusion, solving inequalities can be a challenging task, but by understanding the concept and following the correct procedures, you can arrive at the correct solution. We hope that this Q&A guide has provided a clear and concise explanation of how to solve inequalities and has helped you to understand the concept better.

Final Thoughts

Solving inequalities is an important concept in mathematics, and it has many real-world applications. By understanding how to solve inequalities, you can solve a wide range of problems in fields such as science, engineering, economics, and finance. We hope that this article has provided a useful resource for you to learn and understand the concept of solving inequalities.