Solve The Inequality { -\frac{1}{4} X \ \textless \ 2$}$.A. { X \ \textless \ 8$}$B. { X \ \textgreater \ 8$}$C. { X \ \textgreater \ -8$}$D. { X \ \textless \ -8$}$

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Introduction

Inequalities are mathematical expressions that compare two values using greater than, less than, or equal to symbols. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality −14x<2{-\frac{1}{4} x < 2}. We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is −14x<2{-\frac{1}{4} x < 2}. To solve this inequality, we need to isolate the variable x. The first step is to multiply both sides of the inequality by -1/4. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

Step 1: Multiply Both Sides by -1/4

Multiplying both sides of the inequality by -1/4 gives us:

−14x×−14<2×−14{-\frac{1}{4} x \times -\frac{1}{4} < 2 \times -\frac{1}{4}}

This simplifies to:

x>−8{x > -8}

Step 2: Check the Solution

To check the solution, we can substitute x = -7 into the original inequality:

−14(−7)<2{-\frac{1}{4} (-7) < 2}

Simplifying this expression gives us:

1.75<2{1.75 < 2}

Since this is true, we can conclude that the solution to the inequality is indeed x > -8.

Conclusion

In conclusion, solving the inequality −14x<2{-\frac{1}{4} x < 2} involves multiplying both sides by -1/4 and reversing the direction of the inequality sign. The solution to the inequality is x > -8.

Answer

The correct answer is:

C. x>−8{x > -8}

Tips and Tricks

  • When multiplying or dividing both sides of an inequality by a negative number, remember to reverse the direction of the inequality sign.
  • Always check the solution by substituting a value into the original inequality.
  • Practice solving inequalities to become more comfortable with the process.

Common Mistakes

  • Failing to reverse the direction of the inequality sign when multiplying or dividing by a negative number.
  • Not checking the solution by substituting a value into the original inequality.
  • Not following the order of operations when simplifying expressions.

Real-World Applications

Solving inequalities has many real-world applications, such as:

  • Modeling population growth or decline
  • Determining the maximum or minimum value of a function
  • Finding the optimal solution to a problem

Conclusion

Introduction

In our previous article, we discussed how to solve the inequality −14x<2{-\frac{1}{4} x < 2}. We broke down the solution step by step and provided a clear explanation of each step. In this article, we will answer some common questions related to solving inequalities.

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, on the other hand, is an inequality that can be written in the form ax^2 + bx + c < 0, where a, b, and c are constants.

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. Set each factor equal to zero and solve for x.
  3. Use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses a strict inequality symbol, such as < or >. A non-strict inequality, on the other hand, uses a non-strict inequality symbol, such as ≤ or ≥.

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you can use the following steps:

  1. Graph each inequality on a coordinate plane.
  2. Find the intersection of the two graphs.
  3. Determine the intervals where the system is true.

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

A: A linear programming problem is a problem that involves maximizing or minimizing a linear function subject to a set of linear constraints. A linear inequality, on the other hand, is a mathematical statement that compares two values using a linear expression.

Q: How do I use a graph to solve an inequality?

A: To use a graph to solve an inequality, you can follow these steps:

  1. Graph the inequality on a coordinate plane.
  2. Determine the intervals where the inequality is true.
  3. Use the graph to find the solution to the inequality.

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that involves a rational expression, such as a fraction or a ratio. A polynomial inequality, on the other hand, is an inequality that involves a polynomial expression.

Q: How do I solve a rational inequality?

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

  1. Factor the numerator and denominator, if possible.
  2. Cancel out any common factors.
  3. Use a number line or a graph to determine the intervals where the inequality is true.

Conclusion

Solving inequalities is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can become more comfortable with solving inequalities and apply this skill to a variety of problems.

Tips and Tricks

  • Always check the solution by substituting a value into the original inequality.
  • Use a number line or a graph to determine the intervals where the inequality is true.
  • Practice solving inequalities to become more comfortable with the process.

Common Mistakes

  • Failing to reverse the direction of the inequality sign when multiplying or dividing by a negative number.
  • Not checking the solution by substituting a value into the original inequality.
  • Not following the order of operations when simplifying expressions.

Real-World Applications

Solving inequalities has many real-world applications, such as:

  • Modeling population growth or decline
  • Determining the maximum or minimum value of a function
  • Finding the optimal solution to a problem

Conclusion

Solving inequalities is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can become more comfortable with solving inequalities and apply this skill to a variety of problems.