Solve The Inequality And Write The Solution In Interval Notation.${ 6(4x + 1) \ \textless \ 6 }$Choose The Correct Graph Below.A. B. C. D. Write The Answer In Interval Notation: $ \square $ (Type Your Answer In Interval

Introduction

In this article, we will focus on solving inequalities and writing the solution in interval notation. Inequalities are mathematical statements that compare two expressions using greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves finding the values of the variable that make the inequality true. In this case, we will solve the inequality $6(4x + 1) < 6$ and write the solution in interval notation.

Understanding Interval Notation

Interval notation is a way of writing the solution to an inequality using a specific notation. It consists of a pair of parentheses or brackets that contain the values of the variable that make the inequality true. For example, if we have the inequality $x > 2$, the solution in interval notation would be $(2, \infty)$, which means that $x$ is greater than 2 and can be any value greater than 2.

Solving the Inequality

To solve the inequality $6(4x + 1) < 6$, we need to follow the order of operations (PEMDAS):

1. Distribute the 6: Multiply the 6 to the terms inside the parentheses: $24x + 6 < 6$
2. Subtract 6 from both sides: Subtract 6 from both sides of the inequality to get $24x < 0$
3. Divide both sides by 24: Divide both sides of the inequality by 24 to get $x < 0$

Writing the Solution in Interval Notation

Now that we have solved the inequality, we need to write the solution in interval notation. Since $x < 0$, the solution in interval notation would be $(-\infty, 0)$.

Choosing the Correct Graph

The graph below shows the solution to the inequality $x < 0$. The correct graph is:

A.

Conclusion

In this article, we solved the inequality $6(4x + 1) < 6$ and wrote the solution in interval notation. We also discussed the importance of understanding interval notation and how to choose the correct graph. By following the steps outlined in this article, you should be able to solve inequalities and write the solution in interval notation with ease.

Additional Tips and Tricks

• When solving inequalities, always follow the order of operations (PEMDAS).
• When writing the solution in interval notation, make sure to include the correct notation (parentheses or brackets).
• When choosing the correct graph, make sure to look for the graph that represents the solution to the inequality.

Common Mistakes to Avoid

• Not following the order of operations (PEMDAS) when solving inequalities.
• Not including the correct notation (parentheses or brackets) when writing the solution in interval notation.
• Choosing the incorrect graph when solving inequalities.

Real-World Applications

Solving inequalities and writing the solution in interval notation has many real-world applications. For example, in economics, inequalities are used to model the relationship between variables such as supply and demand. In engineering, inequalities are used to model the behavior of complex systems. In medicine, inequalities are used to model the spread of diseases.

Final Thoughts

Introduction

In our previous article, we discussed how to solve inequalities and write the solution in interval notation. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving inequalities.

Q: What is an inequality?

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

Q: What are the different types of inequalities?

A: There are four main types of inequalities:

• Linear inequalities: Inequalities that can be written in the form $ax + b < c$ or $ax + b > c$, where $a$, $b$, and $c$ are constants.
• Quadratic inequalities: Inequalities that can be written in the form $ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$, where $a$, $b$, and $c$ are constants.
• Polynomial inequalities: Inequalities that can be written in the form $ax^n + bx^{n-1} + \ldots + c < 0$ or $ax^n + bx^{n-1} + \ldots + c > 0$, where $a$, $b$, and $c$ are constants and $n$ is a positive integer.
• Rational inequalities: Inequalities that can be written in the form $\frac{ax + b}{cx + d} < 0$ or $\frac{ax + b}{cx + d} > 0$, where $a$, $b$, $c$, and $d$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, follow these steps:

1. Distribute the inequality: Multiply the inequality to the terms inside the parentheses.
2. Add or subtract the same value to both sides: Add or subtract the same value to both sides of the inequality to get the variable term on one side.
3. Divide both sides by the coefficient: Divide both sides of the inequality by the coefficient of the variable term to get the variable term isolated.
4. Write the solution in interval notation: Write the solution in interval notation using the correct notation (parentheses or brackets).

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, follow these steps:

1. Factor the quadratic expression: Factor the quadratic expression to get the roots of the equation.
2. Use the sign chart method: Use the sign chart method to determine the intervals where the inequality is true.
3. Write the solution in interval notation: Write the solution in interval notation using the correct notation (parentheses or brackets).

Q: What is the difference between a solution set and an interval notation?

A: A solution set is a set of values that satisfy the inequality, while an interval notation is a way of writing the solution set using a specific notation.

Q: How do I choose the correct graph when solving inequalities?

A: To choose the correct graph when solving inequalities, follow these steps:

1. Determine the direction of the inequality: Determine the direction of the inequality (greater than, less than, greater than or equal to, or less than or equal to).
2. Identify the critical points: Identify the critical points (the values that make the inequality true).
3. Plot the critical points: Plot the critical points on the number line.
4. Determine the intervals: Determine the intervals where the inequality is true.
5. Choose the correct graph: Choose the graph that represents the solution to the inequality.

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

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

• Not following the order of operations (PEMDAS): Not following the order of operations (PEMDAS) when solving inequalities.
• Not including the correct notation (parentheses or brackets): Not including the correct notation (parentheses or brackets) when writing the solution in interval notation.
• Choosing the incorrect graph: Choosing the incorrect graph when solving inequalities.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in solving inequalities. We discussed the different types of inequalities, how to solve linear and quadratic inequalities, and how to choose the correct graph when solving inequalities. By following the steps outlined in this article, you should be able to solve inequalities and write the solution in interval notation with ease.