Solve For X X X : 6 \textless 4 X + 8 \textless 14 6 \ \textless \ 4x + 8 \ \textless \ 14 6 \textless 4 X + 8 \textless 14 A. 1 2 \textless X \textless 3 2 \frac{1}{2} \ \textless \ X \ \textless \ \frac{3}{2} 2 1 \textless X \textless 2 3 B. − 2 \textless 4 X \textless 6 -2 \ \textless \ 4x \ \textless \ 6 − 2 \textless 4 X \textless 6 C. $-\frac{1}{2} \ \textless \ X \ \textless

Mar 11, 2025 by ADMIN 404 views

**Solving Inequalities: A Step-by-Step Guide**

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. In this article, we will focus on solving linear inequalities, specifically the inequality $6 < 4x + 8 < 14$ . We will break down the solution into manageable steps, making it easier for readers to understand and apply the concept.

Understanding the Inequality

The given inequality is $6 < 4x + 8 < 14$ . To solve this inequality, we need to isolate the variable $x$ . The first step is to subtract $8$ from all three parts of the inequality, which gives us $-2 < 4x < 6$ .

Subtracting 8 from the Inequality

6 < 4x + 8 < 14
-8 < 4x < 6 - 8
-8 < 4x < -2

Dividing by 4

Now that we have $-2 < 4x < -2$ , we can divide all three parts of the inequality by $4$ . This gives us $-\frac{1}{2} < x < -\frac{1}{2}$ .

-2 < 4x < -2
\frac{-2}{4} < x < \frac{-2}{4}
-\frac{1}{2} < x < -\frac{1}{2}

Simplifying the Inequality

However, we notice that the upper and lower bounds of the inequality are the same, which means that the inequality is not providing any useful information. To simplify the inequality, we can rewrite it as $-\frac{1}{2} < x < -\frac{1}{2}$ .

Conclusion

In conclusion, the solution to the inequality $6 < 4x + 8 < 14$ is $-\frac{1}{2} < x < -\frac{1}{2}$ . However, this solution is not useful, as it does not provide any information about the value of $x$ . To obtain a useful solution, we need to revisit the original inequality and try a different approach.

Alternative Approach

Let's go back to the original inequality $6 < 4x + 8 < 14$ . We can subtract $8$ from all three parts of the inequality, which gives us $-2 < 4x < 6$ . Now, we can divide all three parts of the inequality by $4$ , which gives us $-\frac{1}{2} < x < \frac{3}{2}$ .

6 < 4x + 8 < 14
-8 < 4x < 6 - 8
-8 < 4x < -2
\frac{-8}{4} < x < \frac{-2}{4}
-2 < x < -\frac{1}{2}

Simplifying the Inequality

Now that we have $-\frac{1}{2} < x < \frac{3}{2}$ , we can rewrite the inequality as $\frac{1}{2} < x < \frac{3}{2}$ .

Conclusion

In conclusion, the solution to the inequality $6 < 4x + 8 < 14$ is $\frac{1}{2} < x < \frac{3}{2}$ . This solution provides useful information about the value of $x$ .

Comparison with Other Solutions

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

Option A: $\frac{1}{2} < x < \frac{3}{2}$
Option B: $-2 < 4x < 6$
Option C: $-\frac{1}{2} < x < \frac{3}{2}$

Our solution is the only one that provides a useful range of values for $x$ .

Conclusion

In conclusion, solving inequalities requires careful attention to detail and a step-by-step approach. By following the steps outlined in this article, readers can solve linear inequalities and obtain useful solutions.

Final Answer

Introduction

In our previous article, we discussed how to solve linear inequalities, specifically the inequality $6 < 4x + 8 < 14$ . We broke down the solution into manageable steps and provided a useful solution. In this article, we will answer some common questions related to solving inequalities.

Q: What is an inequality?

A: An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. Inequalities are often used to describe relationships between variables.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

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

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to find the roots of the corresponding quadratic equation. You can do this by factoring the quadratic expression or by using the quadratic formula. Once you have found the roots, you can use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the concept of a number line?

A: A number line is a visual representation of the real numbers, with positive numbers to the right of zero and negative numbers to the left of zero. You can use a number line to determine the intervals where an inequality is true by plotting the roots of the corresponding equation and testing points in each interval.

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

A: To use a number line to solve an inequality, you need to plot the roots of the corresponding equation and test points in each interval. If the point is in the interval where the inequality is true, then the inequality is true for all points in that interval.

Q: What is the concept of a graph?

A: A graph is a visual representation of a function or an inequality. You can use a graph to determine the intervals where an inequality is true by plotting the function or inequality and testing points in each interval.

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

A: To use a graph to solve an inequality, you need to plot the function or inequality and test points in each interval. If the point is in the interval where the inequality is true, then the inequality is true for all points in that interval.

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

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

Not isolating the variable on one side of the inequality sign
Not considering the direction of the inequality sign
Not testing points in each interval
Not using a number line or graph to visualize the solution

Conclusion

Final Answer

The final answer is: There is no final numerical answer to this article. The article provides a Q&A guide to solving inequalities.