Solve The Inequality:$\[ -\frac{a}{4} - 6 + A \neq 9 \\]

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Introduction

In this article, we will delve into the world of inequalities and focus on solving a specific inequality: −a4−6+a≠9-\frac{a}{4} - 6 + a \neq 9. Inequalities are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various mathematical disciplines. In this discussion, we will break down the inequality, identify the steps required to solve it, and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is −a4−6+a≠9-\frac{a}{4} - 6 + a \neq 9. To begin solving this inequality, we need to isolate the variable aa on one side of the inequality. The first step is to simplify the left-hand side of the inequality by combining like terms.

Simplifying the Left-Hand Side

We can start by combining the terms involving aa:

−a4+a=−a4+4a4=3a4-\frac{a}{4} + a = -\frac{a}{4} + \frac{4a}{4} = \frac{3a}{4}

Now, the inequality becomes:

3a4−6≠9\frac{3a}{4} - 6 \neq 9

Adding 6 to Both Sides

To isolate the term involving aa, we need to get rid of the constant term on the left-hand side. We can do this by adding 6 to both sides of the inequality:

3a4≠9+6\frac{3a}{4} \neq 9 + 6

3a4≠15\frac{3a}{4} \neq 15

Multiplying Both Sides by 4

To eliminate the fraction, we can multiply both sides of the inequality by 4:

3a≠603a \neq 60

Solving for aa

Now that we have isolated the term involving aa, we can solve for aa by dividing both sides of the inequality by 3:

a≠603a \neq \frac{60}{3}

a≠20a \neq 20

Conclusion

In this article, we have solved the inequality −a4−6+a≠9-\frac{a}{4} - 6 + a \neq 9 by simplifying the left-hand side, adding 6 to both sides, multiplying both sides by 4, and finally solving for aa. The solution to the inequality is a≠20a \neq 20. This means that the value of aa can be any real number except 20.

Final Answer

The final answer is a≠20\boxed{a \neq 20}.

Frequently Asked Questions

Q: What is the first step in solving the inequality?

A: The first step in solving the inequality is to simplify the left-hand side by combining like terms.

Q: How do we eliminate the fraction in the inequality?

A: We can eliminate the fraction by multiplying both sides of the inequality by the denominator, which is 4 in this case.

Q: What is the solution to the inequality?

A: The solution to the inequality is a≠20a \neq 20.

Step-by-Step Solution

Here is a step-by-step solution to the inequality:

  1. Simplify the left-hand side by combining like terms.
  2. Add 6 to both sides of the inequality.
  3. Multiply both sides of the inequality by 4.
  4. Solve for aa by dividing both sides of the inequality by 3.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid common mistakes such as:

  • Not simplifying the left-hand side of the inequality.
  • Not adding or subtracting the same value from both sides of the inequality.
  • Not multiplying or dividing both sides of the inequality by the same value.

Real-World Applications

Inequalities have numerous real-world applications, including:

  • Finance: Inequalities are used to calculate interest rates, investment returns, and loan payments.
  • Science: Inequalities are used to model population growth, chemical reactions, and physical systems.
  • Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Conclusion

In conclusion, solving the inequality −a4−6+a≠9-\frac{a}{4} - 6 + a \neq 9 requires a step-by-step approach, including simplifying the left-hand side, adding 6 to both sides, multiplying both sides by 4, and finally solving for aa. The solution to the inequality is a≠20a \neq 20. By understanding how to solve inequalities, we can apply this knowledge to real-world problems and make informed decisions.

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Introduction

In our previous article, we solved the inequality −a4−6+a≠9-\frac{a}{4} - 6 + a \neq 9 by simplifying the left-hand side, adding 6 to both sides, multiplying both sides by 4, and finally solving for aa. In this article, we will provide a Q&A guide to help you better understand the solution and address any questions you may have.

Q&A Guide

Q: What is the first step in solving the inequality?

A: The first step in solving the inequality is to simplify the left-hand side by combining like terms.

Q: How do we eliminate the fraction in the inequality?

A: We can eliminate the fraction by multiplying both sides of the inequality by the denominator, which is 4 in this case.

Q: What is the solution to the inequality?

A: The solution to the inequality is a≠20a \neq 20.

Q: Can you explain the concept of inequalities in more detail?

A: Inequalities are mathematical statements that compare two expressions using greater than, less than, or equal to. They can be used to model real-world problems and are an essential part of mathematics.

Q: How do we solve inequalities with fractions?

A: To solve inequalities with fractions, we need to eliminate the fraction by multiplying both sides of the inequality by the denominator.

Q: Can you provide an example of a real-world application of inequalities?

A: Inequalities are used in finance to calculate interest rates, investment returns, and loan payments. For example, if you have a savings account with a 5% interest rate, the inequality a≥1000a \geq 1000 would represent the minimum amount of money you need to deposit to earn at least $50 in interest.

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

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

  • Not simplifying the left-hand side of the inequality.
  • Not adding or subtracting the same value from both sides of the inequality.
  • Not multiplying or dividing both sides of the inequality by the same value.

Q: Can you explain the concept of absolute value in inequalities?

A: Absolute value in inequalities represents the distance between a number and zero. For example, the inequality ∣a∣≥5|a| \geq 5 would represent all values of aa that are greater than or equal to 5 or less than or equal to -5.

Q: How do we solve inequalities with absolute value?

A: To solve inequalities with absolute value, we need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative.

Q: Can you provide an example of a system of inequalities?

A: A system of inequalities is a set of two or more inequalities that must be satisfied simultaneously. For example, the system of inequalities:

a+b≥10a−b≤5\begin{align*} a + b &\geq 10 \\ a - b &\leq 5 \end{align*}

would represent all values of aa and bb that satisfy both inequalities.

Conclusion

In conclusion, solving the inequality −a4−6+a≠9-\frac{a}{4} - 6 + a \neq 9 requires a step-by-step approach, including simplifying the left-hand side, adding 6 to both sides, multiplying both sides by 4, and finally solving for aa. By understanding how to solve inequalities, we can apply this knowledge to real-world problems and make informed decisions. We hope this Q&A guide has helped you better understand the solution and address any questions you may have.

Final Answer

The final answer is a≠20\boxed{a \neq 20}.

Frequently Asked Questions

Q: What is the first step in solving the inequality?

A: The first step in solving the inequality is to simplify the left-hand side by combining like terms.

Q: How do we eliminate the fraction in the inequality?

A: We can eliminate the fraction by multiplying both sides of the inequality by the denominator, which is 4 in this case.

Q: What is the solution to the inequality?

A: The solution to the inequality is a≠20a \neq 20.

Step-by-Step Solution

Here is a step-by-step solution to the inequality:

  1. Simplify the left-hand side by combining like terms.
  2. Add 6 to both sides of the inequality.
  3. Multiply both sides of the inequality by 4.
  4. Solve for aa by dividing both sides of the inequality by 3.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid common mistakes such as:

  • Not simplifying the left-hand side of the inequality.
  • Not adding or subtracting the same value from both sides of the inequality.
  • Not multiplying or dividing both sides of the inequality by the same value.

Real-World Applications

Inequalities have numerous real-world applications, including:

  • Finance: Inequalities are used to calculate interest rates, investment returns, and loan payments.
  • Science: Inequalities are used to model population growth, chemical reactions, and physical systems.
  • Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.