Which Work Correctly Uses Properties Of Inequality To Find The Solution To $-0.4x - 10 \ \textgreater \ 14$?A.$\[ \begin{aligned} -0.4x - 10 & \ \textgreater \ 14 \\ -0.4x & \ \textgreater \ 4 \\ x & \ \textgreater \

Introduction

Inequality is a fundamental concept in mathematics that deals with the comparison of two or more expressions. It is used to represent a relationship between two or more values, and it is a crucial tool in solving equations and inequalities. In this article, we will focus on solving inequalities using the properties of inequality, specifically the given inequality $-0.4x - 10 \ \textgreater \ 14$. We will break down the solution step by step and provide a clear explanation of each step.

Step 1: Add 10 to Both Sides

The first step in solving the inequality is to add 10 to both sides of the equation. This will isolate the term containing the variable, $x$. The inequality becomes:

$$-0.4x - 10 + 10 \ \textgreater \ 14 + 10$$

Simplifying the inequality, we get:

$$-0.4x \ \textgreater \ 24$$

Step 2: Divide Both Sides by -0.4

The next step is to divide both sides of the inequality by -0.4. However, when dividing or multiplying an inequality by a negative number, we must reverse the direction of the inequality sign. The inequality becomes:

$$\frac{-0.4x}{-0.4} \ \textless \ \frac{24}{-0.4}$$

Simplifying the inequality, we get:

$$x \ \textless \ -60$$

Step 3: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. The solution is all values of $x$ that are less than -60. In interval notation, this is written as:

$$x \in (-\infty, -60)$$

Conclusion

In conclusion, to solve the inequality $-0.4x - 10 \ \textgreater \ 14$, we added 10 to both sides, then divided both sides by -0.4, reversing the direction of the inequality sign. The solution is all values of $x$ that are less than -60, written in interval notation as $x \in (-\infty, -60)$.

Discussion

The solution to the inequality $-0.4x - 10 \ \textgreater \ 14$ is a classic example of how to use the properties of inequality to find the solution to a linear inequality. The key steps involved are adding or subtracting the same value to both sides of the inequality, and multiplying or dividing both sides by the same non-zero value, while reversing the direction of the inequality sign when multiplying or dividing by a negative number.

Common Mistakes to Avoid

When solving inequalities, it is essential to remember to reverse the direction of the inequality sign when multiplying or dividing by a negative number. This is a common mistake that can lead to incorrect solutions. Additionally, it is crucial to check the solution by plugging in a value from the solution set into the original inequality to ensure that it is true.

Real-World Applications

Inequalities have numerous real-world applications in fields such as economics, finance, and engineering. For example, in economics, inequalities can be used to model the relationship between supply and demand, while in finance, they can be used to calculate interest rates and investment returns. In engineering, inequalities can be used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

Introduction

In our previous article, we discussed how to solve inequalities using the properties of inequality. 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 statement that compares two or more expressions, indicating that one expression is greater than, less than, or equal to another expression.

Q: What are the properties of inequality?

A: The properties of inequality are:

• Addition Property: If $a > b$, then $a + c > b + c$.
• Subtraction Property: If $a > b$, then $a - c > b - c$.
• Multiplication Property: If $a > b$ and $c > 0$, then $ac > bc$.
• Division Property: If $a > b$ and $c > 0$, then $\frac{a}{c} > \frac{b}{c}$.
• Reversal Property: If $a > b$, then $b < a$.

Q: How do I solve an inequality?

A: To solve an inequality, follow these steps:

1. Add or subtract the same value to both sides of the inequality to isolate the variable.
2. Multiply or divide both sides of the inequality by the same non-zero value to solve for the variable.
3. Reverse the direction of the inequality sign when multiplying or dividing by a negative number.

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 $ax + b > c$ or $ax + b < c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality 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.

Q: How do I graph an inequality?

A: To graph an inequality, follow these steps:

1. Graph the boundary line of the inequality by plotting the points where the inequality is equal to zero.
2. Test a point in each region of the graph to determine which region satisfies the inequality.
3. Shade the region that satisfies the inequality.

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

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

• Not reversing the direction of the inequality sign when multiplying or dividing by a negative number.
• Not checking the solution by plugging in a value from the solution set into the original inequality.
• Not considering the restrictions on the variable, such as the domain of the function.

Conclusion

In conclusion, solving inequalities using the properties of inequality is a crucial skill in mathematics that has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can solve linear and quadratic inequalities with ease and confidence. Remember to add or subtract the same value to both sides of the inequality, multiply or divide both sides by the same non-zero value, and reverse the direction of the inequality sign when multiplying or dividing by a negative number. With practice and patience, you will become proficient in solving inequalities and be able to apply this skill to real-world problems.