What Is The Solution To 3 4 ( X + 8 ) \textgreater 1 2 ( 2 X + 10 \frac{3}{4}(x+8)\ \textgreater \ \frac{1}{2}(2x+10 4 3 ( X + 8 ) \textgreater 2 1 ( 2 X + 10 ]?A. ( − ∞ , − 4 (-\infty, -4 ( − ∞ , − 4 ]B. ( − 4 , ∞ (-4, \infty ( − 4 , ∞ ]C. ( − ∞ , 4 (-\infty, 4 ( − ∞ , 4 ]D. ( 4 , ∞ (4, \infty ( 4 , ∞ ]

Mar 8, 2025 by ADMIN 311 views

$What is the solution to $\frac{3}{4}(x+8)\ \textgreater \ \frac{1}{2}(2x+10$\]?$

Introduction

In this article, we will explore the solution to the given inequality $\frac{3}{4}(x+8)\ \textgreater \ \frac{1}{2}(2x+10$ ]. This type of inequality is a linear inequality, and we will use algebraic methods to solve it. We will start by simplifying the inequality and then isolate the variable x to find the solution.

Step 1: Simplify the Inequality

The first step in solving the inequality is to simplify it by distributing the coefficients to the terms inside the parentheses.

$\frac{3}{4}(x+8)\ \textgreater \ \frac{1}{2}(2x+10$ ]

Distributing the coefficients, we get:

$\frac{3}{4}x + \frac{3}{4}(8)\ \textgreater \ \frac{1}{2}(2x) + \frac{1}{2}(10$ ]

Simplifying further, we get:

$\frac{3}{4}x + 6\ \textgreater \ x + 5$

Step 2: Isolate the Variable x

Now that we have simplified the inequality, we can isolate the variable x by subtracting x from both sides of the inequality.

$\frac{3}{4}x - x + 6\ \textgreater \ x - x + 5$

Simplifying further, we get:

$-\frac{1}{4}x + 6\ \textgreater \ 5$

Step 3: Solve for x

Now that we have isolated the variable x, we can solve for x by subtracting 6 from both sides of the inequality.

$-\frac{1}{4}x + 6 - 6\ \textgreater \ 5 - 6$

Simplifying further, we get:

$-\frac{1}{4}x\ \textgreater \ -1$

Step 4: Multiply Both Sides by -4

To solve for x, we need to multiply both sides of the inequality by -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.

$-1 \cdot -\frac{1}{4}x\ \textless \ -1 \cdot -1$

Simplifying further, we get:

$x\ \textless \ 4$

Conclusion

The solution to the inequality $\frac{3}{4}(x+8)\ \textgreater \ \frac{1}{2}(2x+10$ ] is x < 4. This means that the value of x must be less than 4 to satisfy the inequality.

Final Answer

The final answer is x < 4, which corresponds to option C. $(-\infty, 4$ ].

Discussion

The solution to the inequality $\frac{3}{4}(x+8)\ \textgreater \ \frac{1}{2}(2x+10$ ] is x < 4. This means that the value of x must be less than 4 to satisfy the inequality. The correct answer is option C. $(-\infty, 4$ ].

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart
[3] "Inequalities" by Michael Artin

Introduction

In the previous article, we explored the solution to the inequality $\frac{3}{4}(x+8)\ \textgreater \ \frac{1}{2}(2x+10$ ]. In this article, we will answer some frequently asked questions (FAQs) on solving linear inequalities.

Q1: What is a linear inequality?

A linear inequality is an inequality that can be written in the form ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c, where a, b, and c are constants, and x is the variable.

Q2: How do I solve a linear inequality?

To solve a linear inequality, you need to isolate the variable x by performing algebraic operations such as addition, subtraction, multiplication, and division. You also need to reverse the direction of the inequality when multiplying or dividing both sides by a negative number.

Q3: What is the difference between a linear inequality and a linear equation?

A linear equation is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c.

Q4: How do I determine the direction of the inequality?

When solving a linear inequality, you need to determine the direction of the inequality. If you multiply or divide both sides of the inequality by a positive number, the direction of the inequality remains the same. However, if you multiply or divide both sides of the inequality by a negative number, you need to reverse the direction of the inequality.

Q5: What is the solution to a linear inequality?

The solution to a linear inequality is the set of all values of x that satisfy the inequality. For example, if the inequality is x > 2, the solution is all values of x that are greater than 2.

Q6: How do I graph a linear inequality?

To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality. If the inequality is of the form x > a, you need to shade the region to the right of the line x = a. If the inequality is of the form x < a, you need to shade the region to the left of the line x = a.

Q7: What is the difference between a linear inequality and a quadratic inequality?

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

Q8: How do I solve a quadratic inequality?

To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution. You can also use the quadratic formula to find the solutions to the quadratic equation and then use the sign of the quadratic expression to determine the solution.

Q9: What is the importance of solving linear inequalities?

Solving linear inequalities is important in many real-world applications, such as finance, economics, and engineering. For example, in finance, linear inequalities are used to determine the minimum or maximum value of an investment. In economics, linear inequalities are used to determine the optimal level of production or consumption.

Q10: How can I practice solving linear inequalities?

You can practice solving linear inequalities by working on problems and exercises in a textbook or online resource. You can also use online tools and software to practice solving linear inequalities.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) on solving linear inequalities. We have discussed the definition of a linear inequality, how to solve a linear inequality, and the importance of solving linear inequalities. We have also provided some tips and resources for practicing solving linear inequalities.

Final Answer

The final answer is that solving linear inequalities is an important skill that has many real-world applications. By practicing solving linear inequalities, you can develop your problem-solving skills and become more confident in your ability to solve complex problems.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart
[3] "Inequalities" by Michael Artin

Introduction

Step 1: Simplify the Inequality

Step 2: Isolate the Variable x

Step 3: Solve for x

Step 4: Multiply Both Sides by -4

Conclusion

Final Answer

Discussion

Related Topics

References

Introduction

Q1: What is a linear inequality?

Q2: How do I solve a linear inequality?

Q3: What is the difference between a linear inequality and a linear equation?

Q4: How do I determine the direction of the inequality?

Q5: What is the solution to a linear inequality?

Q6: How do I graph a linear inequality?

Q7: What is the difference between a linear inequality and a quadratic inequality?

Q8: How do I solve a quadratic inequality?

Q9: What is the importance of solving linear inequalities?

Q10: How can I practice solving linear inequalities?

Conclusion

Final Answer

Related Topics

References