Solve The Inequality:$\[ -6(1 + 6x) \ \textless \ 6(1 - 5x) \\]

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Introduction

In this article, we will delve into the world of inequalities and explore a specific problem that requires us to solve an inequality involving variables. The given inequality is −6(1+6x) \textless 6(1−5x)-6(1 + 6x) \ \textless \ 6(1 - 5x). Our goal is to isolate the variable and determine the range of values that satisfy the inequality.

Understanding the Inequality

Before we begin solving the inequality, let's first understand what it means. The inequality −6(1+6x) \textless 6(1−5x)-6(1 + 6x) \ \textless \ 6(1 - 5x) can be read as "negative six times one plus six x is less than six times one minus five x." This means that the expression on the left-hand side is less than the expression on the right-hand side.

Distributing the Negative Sign

To simplify the inequality, we can start by distributing the negative sign to the terms inside the parentheses on the left-hand side. This gives us:

−6(1+6x)=−6−36x-6(1 + 6x) = -6 - 36x

Distributing the Positive Sign

Next, we can distribute the positive sign to the terms inside the parentheses on the right-hand side. This gives us:

6(1−5x)=6−30x6(1 - 5x) = 6 - 30x

Combining Like Terms

Now that we have distributed the signs, we can combine like terms on both sides of the inequality. This gives us:

−6−36x \textless 6−30x-6 - 36x \ \textless \ 6 - 30x

Adding 36x to Both Sides

To isolate the variable, we can add 36x to both sides of the inequality. This gives us:

−6 \textless 6−6x-6 \ \textless \ 6 - 6x

Adding 6 to Both Sides

Next, we can add 6 to both sides of the inequality. This gives us:

0 \textless 12−6x0 \ \textless \ 12 - 6x

Subtracting 12 from Both Sides

To further isolate the variable, we can subtract 12 from both sides of the inequality. This gives us:

−12 \textless −6x-12 \ \textless \ -6x

Dividing Both Sides by -6

Finally, we can divide both sides of the inequality by -6. However, since we are dividing by a negative number, we need to reverse the direction of the inequality. This gives us:

2 \textgreater x2 \ \textgreater \ x

Conclusion

In conclusion, the solution to the inequality −6(1+6x) \textless 6(1−5x)-6(1 + 6x) \ \textless \ 6(1 - 5x) is x \textless 2x \ \textless \ 2. This means that the value of x must be less than 2 in order to satisfy the inequality.

Example Use Case

Suppose we have a situation where we need to determine the range of values for x that satisfy the inequality. For example, if we have a function f(x) = 2x + 1, and we want to find the values of x for which f(x) is less than 5, we can use the solution to the inequality to determine the range of values for x.

Tips and Tricks

When solving inequalities, it's essential to remember to distribute the signs correctly and to combine like terms. Additionally, when dividing by a negative number, we need to reverse the direction of the inequality.

Common Mistakes

One common mistake when solving inequalities is to forget to distribute the signs correctly. Another mistake is to forget to reverse the direction of the inequality when dividing by a negative number.

Final Thoughts

Solving inequalities can be a challenging task, but with practice and patience, it can become second nature. Remember to distribute the signs correctly, combine like terms, and reverse the direction of the inequality when dividing by a negative number. With these tips and tricks, you'll be well on your way to becoming a master of inequality solving.

Frequently Asked Questions

Q: What is the solution to the inequality −6(1+6x) \textless 6(1−5x)-6(1 + 6x) \ \textless \ 6(1 - 5x)?

A: The solution to the inequality is x \textless 2x \ \textless \ 2.

Q: How do I distribute the signs correctly when solving inequalities?

A: To distribute the signs correctly, you need to multiply the negative sign by each term inside the parentheses.

Q: What happens when I divide by a negative number when solving inequalities?

A: When you divide by a negative number, you need to reverse the direction of the inequality.

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

A: Some common mistakes to avoid include forgetting to distribute the signs correctly and forgetting to reverse the direction of the inequality when dividing by a negative number.

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through example problems and exercises. You can also try solving inequalities with different variables and coefficients to get a feel for how the solution changes.

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Introduction

In our previous article, we explored the world of inequalities and solved a specific problem involving variables. In this article, we will continue to delve into the world of inequalities and provide a Q&A guide to help you better understand and solve inequalities.

Q&A: Solving Inequalities

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two expressions using a relation such as greater than, less than, greater than or equal to, or less than or equal to.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing operations that do not change the direction of the inequality. This may involve adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: What is the difference between solving an equation and solving an inequality?

A: Solving an equation involves finding the value of the variable that makes the equation true, while solving an inequality involves finding the range of values of the variable that satisfy the inequality.

Q: How do I know when to reverse the direction of the inequality?

A: You need to reverse the direction of the inequality when dividing both sides of the inequality by a negative number.

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

A: Some common mistakes to avoid include forgetting to distribute the signs correctly, forgetting to combine like terms, and forgetting to reverse the direction of the inequality when dividing by a negative number.

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through example problems and exercises. You can also try solving inequalities with different variables and coefficients to get a feel for how the solution changes.

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

A: Solving inequalities has many real-world applications, including finance, economics, and engineering. For example, you may need to solve an inequality to determine the range of values for a variable that satisfies a certain condition.

Q: Can I use algebraic properties to solve inequalities?

A: Yes, you can use algebraic properties such as the distributive property, the commutative property, and the associative property to solve inequalities.

Q: How do I know when to use the distributive property to solve an inequality?

A: You need to use the distributive property when you have an inequality with a product of two or more terms on one side of the inequality.

Q: What are some tips for solving inequalities?

A: Some tips for solving inequalities include:

  • Start by simplifying the inequality by combining like terms and distributing the signs correctly.
  • Use algebraic properties such as the distributive property and the commutative property to simplify the inequality.
  • Reverse the direction of the inequality when dividing both sides of the inequality by a negative number.
  • Check your solution by plugging it back into the original inequality.

Conclusion

Solving inequalities can be a challenging task, but with practice and patience, it can become second nature. By following the tips and tricks outlined in this article, you can improve your skills and become a master of inequality solving.

Example Use Case

Suppose we have a situation where we need to determine the range of values for x that satisfy the inequality 2x + 5 > 10. We can use the solution to the inequality to determine the range of values for x.

Tips and Tricks

When solving inequalities, it's essential to remember to distribute the signs correctly and to combine like terms. Additionally, when dividing by a negative number, we need to reverse the direction of the inequality.

Common Mistakes

One common mistake when solving inequalities is to forget to distribute the signs correctly. Another mistake is to forget to reverse the direction of the inequality when dividing by a negative number.

Final Thoughts

Solving inequalities can be a challenging task, but with practice and patience, it can become second nature. Remember to distribute the signs correctly, combine like terms, and reverse the direction of the inequality when dividing by a negative number. With these tips and tricks, you'll be well on your way to becoming a master of inequality solving.

Frequently Asked Questions

Q: What is the solution to the inequality 2x + 5 > 10?

A: The solution to the inequality is x > 2.5.

Q: How do I distribute the signs correctly when solving inequalities?

A: To distribute the signs correctly, you need to multiply the negative sign by each term inside the parentheses.

Q: What happens when I divide by a negative number when solving inequalities?

A: When you divide by a negative number, you need to reverse the direction of the inequality.

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

A: Some common mistakes to avoid include forgetting to distribute the signs correctly and forgetting to reverse the direction of the inequality when dividing by a negative number.

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through example problems and exercises. You can also try solving inequalities with different variables and coefficients to get a feel for how the solution changes.