Solve The Inequality:$\[ -\frac{1}{4}(w-5) \geq -2 \\]

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. Inequalities are mathematical statements that compare two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving linear inequalities, specifically the inequality $-\frac{1}{4}(w-5) \geq -2$. We will break down the solution step by step, providing a clear understanding of the concept and the necessary steps to solve it.

What are Linear Inequalities?

Linear inequalities are mathematical statements that compare two linear expressions, indicating that one is greater than, less than, or equal to the other. They are represented in the form of $ax + b \geq c$ or $ax + b \leq c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Linear inequalities can be solved using various methods, including algebraic manipulation, graphical representation, and numerical methods.

The Given Inequality

The given inequality is $-\frac{1}{4}(w-5) \geq -2$. To solve this inequality, we need to isolate the variable $w$ on one side of the inequality sign.

Step 1: Multiply Both Sides by -4

To eliminate the fraction, we can multiply both sides of the inequality by $-4$. This will give us:

$$-(w-5) \geq -8$$

Step 2: Distribute the Negative Sign

When we multiply a negative number by a binomial, we need to distribute the negative sign to both terms inside the parentheses. This will give us:

$$-w + 5 \geq -8$$

Step 3: Add 8 to Both Sides

To isolate the variable $w$, we need to add 8 to both sides of the inequality. This will give us:

$$-w + 13 \geq 0$$

Step 4: Subtract 13 from Both Sides

To isolate the variable $w$, we need to subtract 13 from both sides of the inequality. This will give us:

$$-w \geq -13$$

Step 5: Multiply Both Sides by -1

To isolate the variable $w$, we need to multiply both sides of the inequality by $-1$. This will give us:

$$w \leq 13$$

Conclusion

In this article, we solved the inequality $-\frac{1}{4}(w-5) \geq -2$ step by step. We used algebraic manipulation to isolate the variable $w$ on one side of the inequality sign. The final solution is $w \leq 13$. This means that the value of $w$ must be less than or equal to 13 to satisfy the given inequality.

Graphical Representation

To visualize the solution, we can graph the inequality on a number line. The number line represents the possible values of $w$. The inequality $w \leq 13$ indicates that the value of $w$ must be less than or equal to 13. This can be represented on a number line as follows:

The number line shows that the value of $w$ must be less than or equal to 13. This is represented by the closed circle at 13.

Real-World Applications

Linear inequalities have numerous real-world applications. They are used in various fields, including economics, finance, and engineering. For example, in economics, linear inequalities can be used to model the relationship between two variables, such as the demand for a product and its price. In finance, linear inequalities can be used to model the relationship between the return on investment and the risk level. In engineering, linear inequalities can be used to model the relationship between the stress and strain on a material.

Conclusion

Introduction

In our previous article, we solved the inequality $-\frac{1}{4}(w-5) \geq -2$ step by step. We used algebraic manipulation to isolate the variable $w$ on one side of the inequality sign. The final solution is $w \leq 13$. In this article, we will provide a Q&A guide to help you understand and solve linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is a mathematical statement that compares two linear expressions, indicating that one is greater than, less than, or equal to the other. They are represented in the form of $ax + b \geq c$ or $ax + b \leq c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is a mathematical statement that states that two linear expressions are equal. A linear inequality, on the other hand, states that one linear expression is greater than, less than, or equal to another linear expression.

Q: Can I use the same methods to solve linear inequalities as I do to solve linear equations?

A: Yes, you can use the same methods to solve linear inequalities as you do to solve linear equations. However, you need to be careful when multiplying or dividing both sides of the inequality by a negative value, as this can change the direction of the inequality.

Q: How do I graph a linear inequality on a number line?

A: To graph a linear inequality on a number line, you need to identify the values of the variable that satisfy the inequality. If the inequality is of the form $x \geq a$, you can graph a closed circle at $a$. If the inequality is of the form $x \leq a$, you can graph a closed circle at $a$. If the inequality is of the form $x > a$, you can graph an open circle at $a$. If the inequality is of the form $x < a$, you can graph an open circle at $a$.

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

A: Linear inequalities have numerous real-world applications. They are used in various fields, including economics, finance, and engineering. For example, in economics, linear inequalities can be used to model the relationship between two variables, such as the demand for a product and its price. In finance, linear inequalities can be used to model the relationship between the return on investment and the risk level. In engineering, linear inequalities can be used to model the relationship between the stress and strain on a material.

Q: Can I use technology to solve linear inequalities?

A: Yes, you can use technology to solve linear inequalities. Many graphing calculators and computer algebra systems can be used to solve linear inequalities. Additionally, many online resources and software programs are available to help you solve linear inequalities.

Conclusion

In conclusion, solving linear inequalities is an essential skill in mathematics. It requires a clear understanding of the concept and the necessary steps to solve it. In this article, we provided a Q&A guide to help you understand and solve linear inequalities. We hope that this guide has been helpful in answering your questions and providing you with a better understanding of linear inequalities.