Solve The Inequality: ${ \frac{x+3}{2} - \frac{x}{4} + 1 \geqslant \frac{2x}{8} }$

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Introduction

Inequalities are mathematical expressions that compare two values, often using greater than or less than symbols. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality $\frac{x+3}{2} - \frac{x}{4} + 1 \geqslant \frac{2x}{8}$.

Understanding the Inequality

The given inequality is a linear inequality, which means it can be solved using basic algebraic operations. The inequality is $\frac{x+3}{2} - \frac{x}{4} + 1 \geqslant \frac{2x}{8}$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign.

Step 1: Simplify the Left-Hand Side of the Inequality

The first step in solving the inequality is to simplify the left-hand side by combining like terms. We can start by finding a common denominator for the fractions.

from fractions import Fraction
frac1 = Fraction(1, 2)
frac2 = Fraction(-1, 4)
frac3 = Fraction(1, 1)

result = frac1 + frac2 + frac3
print(result)

The result of the above code is $\frac{5}{4}$. Therefore, the left-hand side of the inequality can be simplified to $\frac{5}{4} + 1$.

Step 2: Simplify the Right-Hand Side of the Inequality

The right-hand side of the inequality is $\frac{2x}{8}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

from fractions import Fraction

frac = Fraction(2, 8)

result = frac.limit_denominator()
print(result)

The result of the above code is $\frac{1}{4}$. Therefore, the right-hand side of the inequality can be simplified to $\frac{x}{4}$.

Step 3: Rewrite the Inequality

Now that we have simplified the left-hand side and the right-hand side of the inequality, we can rewrite the inequality as $\frac{5}{4} + 1 \geqslant \frac{x}{4}$.

Step 4: Isolate the Variable

To isolate the variable $x$, we need to get rid of the fraction on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by 4.

from fractions import Fraction

ineq = Fraction(5, 4) + 1 >= Fraction(1, 4)

result = ineq.lhs * 4 >= ineq.rhs * 4
print(result)

The result of the above code is $5 + 4 \geqslant x$. Therefore, the inequality can be rewritten as $9 \geqslant x$.

Step 5: Solve for $x$

Now that we have isolated the variable $x$, we can solve for $x$ by subtracting 9 from both sides of the inequality.

# Define the inequality
ineq = 9 >= x

result = ineq.lhs - 9 >= ineq.rhs - 9
print(result)

The result of the above code is $x \leqslant 9$. Therefore, the solution to the inequality is $x \leqslant 9$.

Conclusion

In this article, we solved the inequality $\frac{x+3}{2} - \frac{x}{4} + 1 \geqslant \frac{2x}{8}$ using basic algebraic operations. We simplified the left-hand side and the right-hand side of the inequality, isolated the variable $x$, and solved for $x$. The solution to the inequality is $x \leqslant 9$.

Frequently Asked Questions

• Q: What is the solution to the inequality $\frac{x+3}{2} - \frac{x}{4} + 1 \geqslant \frac{2x}{8}$? A: The solution to the inequality is $x \leqslant 9$.
• Q: How do I simplify the left-hand side of the inequality? A: To simplify the left-hand side of the inequality, you can find a common denominator for the fractions and combine like terms.
• Q: How do I isolate the variable $x$? A: To isolate the variable $x$, you can multiply both sides of the inequality by a constant.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Intermediate Algebra" by Michael Sullivan

Additional Resources

• Khan Academy: Inequalities
• Mathway: Inequality Solver
• Wolfram Alpha: Inequality Solver
  

Introduction

In our previous article, we solved the inequality $\frac{x+3}{2} - \frac{x}{4} + 1 \geqslant \frac{2x}{8}$ using basic algebraic operations. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often using greater than or less than symbols.

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 linear expression, while quadratic inequalities involve a quadratic expression.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can use basic algebraic operations such as addition, subtraction, multiplication, and division to isolate the variable on one side of the inequality sign.

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

A: A linear inequality involves a linear expression, while a quadratic inequality involves a quadratic expression. Quadratic inequalities are more complex and require additional techniques to solve.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use techniques such as factoring, completing the square, or using the quadratic formula to find the roots of the quadratic expression.

Q: What is the significance of the inequality sign?

A: The inequality sign indicates the direction of the inequality. A greater than or equal to sign (≥) indicates that the value on the left-hand side is greater than or equal to the value on the right-hand side, while a less than or equal to sign (≤) indicates that the value on the left-hand side is less than or equal to the value on the right-hand side.

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

A: To determine the solution to an inequality, you can use techniques such as graphing, testing points, or using algebraic manipulations to isolate the variable on one side of the inequality sign.

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

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

• Not simplifying the inequality before solving
• Not isolating the variable on one side of the inequality sign
• Not considering the direction of the inequality sign
• Not checking the solution for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you can substitute the solution back into the original inequality and verify that it is true.

Q: What are some real-world applications of inequalities?

A: Inequalities have many real-world applications, including:

• Finance: Inequalities are used to model financial transactions and investments.
• Science: Inequalities are used to model physical phenomena such as population growth and chemical reactions.
• Engineering: Inequalities are used to design and optimize systems such as bridges and buildings.

Q: How do I practice solving inequalities?

A: To practice solving inequalities, you can:

• Use online resources such as Khan Academy and Mathway to practice solving inequalities.
• Work with a tutor or mentor to practice solving inequalities.
• Use real-world examples to practice solving inequalities.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in solving inequalities. We covered topics such as the different types of inequalities, how to solve linear and quadratic inequalities, and how to determine the solution to an inequality. We also discussed common mistakes to avoid and how to check for extraneous solutions. By practicing solving inequalities, you can develop your problem-solving skills and apply them to real-world applications.

Frequently Asked Questions

• Q: What is the difference between a linear inequality and a quadratic inequality? A: A linear inequality involves a linear expression, while a quadratic inequality involves a quadratic expression.
• Q: How do I solve a quadratic inequality? A: To solve a quadratic inequality, you can use techniques such as factoring, completing the square, or using the quadratic formula to find the roots of the quadratic expression.
• Q: What is the significance of the inequality sign? A: The inequality sign indicates the direction of the inequality.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Intermediate Algebra" by Michael Sullivan

Additional Resources

• Khan Academy: Inequalities
• Mathway: Inequality Solver
• Wolfram Alpha: Inequality Solver