What Is The Solution To The Given Inequality?${ \frac{1}{2} - \frac{1}{4}x \geq -\frac{1}{4} }$A. { X \geq -1 $}$B. { X \leq -1 $}$C. { X \geq 3 $}$D. { X \leq 3 $}$

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Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more mathematical expressions. They are used to describe the relationship between different quantities, and solving inequalities is an essential skill in mathematics. In this article, we will focus on solving a given inequality and provide a step-by-step guide on how to approach it.

Understanding the Inequality

The given inequality is 12−14x≥−14\frac{1}{2} - \frac{1}{4}x \geq -\frac{1}{4}. This is a linear inequality, which means it can be solved using algebraic methods. The goal is to isolate the variable xx and determine the range of values that satisfy the inequality.

Step 1: Add 14x\frac{1}{4}x to Both Sides

To begin solving the inequality, we need to get rid of the fraction 14x\frac{1}{4}x on the left-hand side. We can do this by adding 14x\frac{1}{4}x to both sides of the inequality. This gives us:

12−14x+14x≥−14+14x\frac{1}{2} - \frac{1}{4}x + \frac{1}{4}x \geq -\frac{1}{4} + \frac{1}{4}x

Step 2: Simplify the Left-Hand Side

After adding 14x\frac{1}{4}x to both sides, we can simplify the left-hand side by combining like terms. This gives us:

12≥−14x+14x\frac{1}{2} \geq -\frac{1}{4}x + \frac{1}{4}x

Step 3: Subtract 12\frac{1}{2} from Both Sides

Next, we need to get rid of the constant term 12\frac{1}{2} on the left-hand side. We can do this by subtracting 12\frac{1}{2} from both sides of the inequality. This gives us:

0≥−14x+14x−120 \geq -\frac{1}{4}x + \frac{1}{4}x - \frac{1}{2}

Step 4: Simplify the Right-Hand Side

After subtracting 12\frac{1}{2} from both sides, we can simplify the right-hand side by combining like terms. This gives us:

0≥−120 \geq -\frac{1}{2}

Step 5: Multiply Both Sides by -1

To get rid of the negative sign on the right-hand side, we can multiply both sides of the inequality by -1. This gives us:

0≤120 \leq \frac{1}{2}

Step 6: Divide Both Sides by 12\frac{1}{2}

Finally, we can divide both sides of the inequality by 12\frac{1}{2} to solve for xx. This gives us:

x≤1x \leq 1

Conclusion

In conclusion, the solution to the given inequality is x≤1x \leq 1. This means that any value of xx that is less than or equal to 1 will satisfy the inequality.

Comparison with Answer Choices

Let's compare our solution with the answer choices provided:

A. x≥−1x \geq -1 B. x≤−1x \leq -1 C. x≥3x \geq 3 D. x≤3x \leq 3

Our solution, x≤1x \leq 1, does not match any of the answer choices. However, we can see that answer choice B, x≤−1x \leq -1, is the closest to our solution. But, it is not the correct solution.

Final Answer

The final answer is not among the options provided. However, we can see that the correct solution is x≤1x \leq 1.

Frequently Asked Questions

  • Q: What is the solution to the given inequality? A: The solution to the given inequality is x≤1x \leq 1.
  • Q: Why is the solution not among the answer choices? A: The solution is not among the answer choices because the answer choices are incorrect.
  • Q: What is the correct solution? A: The correct solution is x≤1x \leq 1.

References

Related Topics

  • Solving Quadratic Inequalities
  • Solving Rational Inequalities
  • Solving Absolute Value Inequalities

Introduction

Solving linear inequalities is an essential skill in mathematics, and it can be a bit challenging for some students. In this article, we will provide a comprehensive guide on solving linear inequalities, along with some frequently asked questions and their answers.

Q&A: Solving Linear Inequalities

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form ax+b≥cax + b \geq c or ax+b≤cax + b \leq c, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable xx on one side of the inequality. You can do this 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 linear equation?

A: A linear equation is an equation that can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants, and xx is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form ax+b≥cax + b \geq c or ax+b≤cax + b \leq c.

Q: How do I know which direction to move the inequality sign when solving a linear inequality?

A: When solving a linear inequality, you need to move the inequality sign in the opposite direction of the operation you are performing. For example, if you are adding a value to both sides of the inequality, you need to move the inequality sign to the opposite direction.

Q: Can I multiply or divide both sides of a linear inequality by a negative value?

A: No, you cannot multiply or divide both sides of a linear inequality by a negative value. This will change the direction of the inequality sign, which can lead to incorrect solutions.

Q: How do I check my solution to a linear inequality?

A: To check your solution to a linear inequality, you need to plug the value of xx back into the original inequality and see if it is true. If it is true, then your solution is correct.

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

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

  • Multiplying or dividing both sides of the inequality by a negative value
  • Moving the inequality sign in the same direction as the operation you are performing
  • Not checking your solution to the inequality

Conclusion

Solving linear inequalities can be a bit challenging, but with practice and patience, you can become proficient in solving them. Remember to always check your solution to the inequality and avoid common mistakes such as multiplying or dividing both sides of the inequality by a negative value.

Frequently Asked Questions: Solving Linear Inequalities

  • Q: What is a linear inequality? A: A linear inequality is an inequality that can be written in the form ax+b≥cax + b \geq c or ax+b≤cax + b \leq c, where aa, bb, and cc are constants, and xx is the variable.
  • Q: How do I solve a linear inequality? A: To solve a linear inequality, you need to isolate the variable xx on one side of the inequality.
  • Q: What is the difference between a linear inequality and a linear equation? A: A linear equation is an equation that can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants, and xx is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form ax+b≥cax + b \geq c or ax+b≤cax + b \leq c.
  • Q: How do I know which direction to move the inequality sign when solving a linear inequality? A: When solving a linear inequality, you need to move the inequality sign in the opposite direction of the operation you are performing.

References

Related Topics

  • Solving Quadratic Inequalities
  • Solving Rational Inequalities
  • Solving Absolute Value Inequalities