Which Inequality Represents All The Solutions Of \[$8(6x - 7) \ \textless \ 5(9x - 4)\$\]?A. \[$x \ \textgreater \ 12\$\] B. \[$x \ \textless \ 12\$\] C. \[$x \ \textgreater \ 20\$\] D. \[$x \ \textless \

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Introduction to Inequalities

Inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. In this article, we will focus on solving inequalities and understanding the solutions to a given inequality expression. We will use the inequality ${8(6x - 7) \ \textless \ 5(9x - 4)\$} as an example to demonstrate the step-by-step process of solving inequalities.

Understanding the Given Inequality

The given inequality is ${8(6x - 7) \ \textless \ 5(9x - 4)\$}. To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expression.

Simplifying the Inequality

To simplify the inequality, we need to distribute the numbers outside the parentheses to the terms inside.

${8(6x - 7) \ \textless \ 5(9x - 4)\$}

Using the distributive property, we get:

${48x - 56 \ \textless \ 45x - 20\$}

Combining Like Terms

Now, we need to combine like terms on both sides of the inequality.

${48x - 56 \ \textless \ 45x - 20\$}

Subtracting 45x from both sides gives us:

${3x - 56 \ \textless \ -20\$}

Adding 56 to Both Sides

To isolate the term with the variable, we need to add 56 to both sides of the inequality.

${3x - 56 \ \textless \ -20\$}

Adding 56 to both sides gives us:

${3x \ \textless \ 36\$}

Dividing Both Sides by 3

Finally, we need to divide both sides of the inequality by 3 to solve for x.

${3x \ \textless \ 36\$}

Dividing both sides by 3 gives us:

{x \ \textless \ 12$}$

Conclusion

In this article, we solved the inequality ${8(6x - 7) \ \textless \ 5(9x - 4)\$} and found that the solution is {x \ \textless \ 12$}$. This means that all values of x that are less than 12 satisfy the given inequality.

Which Inequality Represents All the Solutions?

Now that we have solved the inequality, we can compare our solution to the given options.

A. {x \ \textgreater \ 12$}$ B. {x \ \textless \ 12$}$ C. {x \ \textgreater \ 20$}$ D. {x \ \textless \ 20$}$

Based on our solution, we can see that option B is the correct answer.

Final Answer

The final answer is option B: {x \ \textless \ 12$}$.

Introduction

In the previous article, we solved the inequality ${8(6x - 7) \ \textless \ 5(9x - 4)\$} and found that the solution is {x \ \textless \ 12$}$. In this article, we will answer some frequently asked questions (FAQs) about solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, indicating whether one value is greater than, less than, or equal to another value.

Q: How do I solve an inequality?

A: To solve an inequality, you need to follow the order of operations (PEMDAS) and simplify the expression. You can use the distributive property to distribute numbers outside the parentheses to the terms inside. Then, you can combine like terms on both sides of the inequality and isolate the term with the variable.

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 \ \textless \ cx + d$}$, where a, b, c, and d are constants. A quadratic inequality is an inequality that can be written in the form {ax^2 + bx + c \ \textless \ dx^2 + ex + f$}$, where a, b, c, d, e, and f are constants.

Q: How do I graph an inequality?

A: To graph an inequality, you need to find the boundary line and then test a point on either side of the line to determine which side of the line satisfies the inequality.

Q: What is the solution to an inequality?

A: The solution to an inequality is the set of all values that satisfy the inequality. In other words, it is the set of all values that make the inequality true.

Q: Can I have multiple solutions to an inequality?

A: Yes, it is possible to have multiple solutions to an inequality. For example, the inequality {x \ \textless \ 2$}$ has multiple solutions, including all values less than 2.

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

A: To determine the direction of the inequality, you need to look at the sign of the coefficient of the variable. If the coefficient is positive, the inequality is greater than or equal to. If the coefficient is negative, the inequality is less than or equal to.

Q: Can I have a compound inequality?

A: Yes, it is possible to have a compound inequality, which is an inequality that involves two or more inequalities joined by the word "and" or "or".

Conclusion

In this article, we answered some frequently asked questions (FAQs) about solving inequalities. We hope that this article has provided you with a better understanding of how to solve inequalities and has helped you to become more confident in your ability to solve them.

Final Tips

  • Always follow the order of operations (PEMDAS) when solving an inequality.
  • Use the distributive property to distribute numbers outside the parentheses to the terms inside.
  • Combine like terms on both sides of the inequality and isolate the term with the variable.
  • Graph the inequality by finding the boundary line and testing a point on either side of the line.
  • Determine the direction of the inequality by looking at the sign of the coefficient of the variable.

We hope that this article has been helpful in your understanding of solving inequalities. If you have any further questions or need additional help, please don't hesitate to ask.