Which Inequality Represents All The Solutions Of − 5 X + 5 ≥ 160 − 10 X -5x + 5 \geq 160 - 10x − 5 X + 5 ≥ 160 − 10 X ?A. X ≤ − 33 X \leq -33 X ≤ − 33 B. X ≥ 33 X \geq 33 X ≥ 33 C. X ≥ 31 X \geq 31 X ≥ 31 D. X ≤ 31 X \leq 31 X ≤ 31

Introduction

In mathematics, inequalities are used to represent relationships between variables. Solving linear inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $-5x + 5 \geq 160 - 10x$ and determine which of the given options represents all the solutions.

Understanding the Inequality

The given inequality is $-5x + 5 \geq 160 - 10x$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality. We can start by adding $10x$ to both sides of the inequality to get:

$$-5x + 10x + 5 \geq 160$$

This simplifies to:

$$5x + 5 \geq 160$$

Isolating the Variable

Next, we need to isolate the variable $x$ on one side of the inequality. We can do this by subtracting $5$ from both sides of the inequality to get:

$$5x \geq 155$$

Solving for x

Now that we have isolated the variable $x$, we can solve for $x$ by dividing both sides of the inequality by $5$:

$$x \geq \frac{155}{5}$$

This simplifies to:

$$x \geq 31$$

Analyzing the Options

Now that we have solved the inequality, we can analyze the given options to determine which one represents all the solutions.

• Option A: $x \leq -33$
• Option B: $x \geq 33$
• Option C: $x \geq 31$
• Option D: $x \leq 31$

Based on our solution, we can see that the correct option is:

Option C: $x \geq 31$

This option represents all the solutions of the inequality $-5x + 5 \geq 160 - 10x$.

Conclusion

In this article, we solved the linear inequality $-5x + 5 \geq 160 - 10x$ and determined which of the given options represents all the solutions. We used basic algebraic operations to isolate the variable $x$ and solve for its value. The correct option is $x \geq 31$, which represents all the solutions of the given inequality.

Tips and Tricks

When solving linear inequalities, it's essential to follow the order of operations and isolate the variable on one side of the inequality. Additionally, be careful when dividing or multiplying both sides of the inequality by a negative number, as this can change the direction of the inequality.

Common Mistakes

When solving linear inequalities, some common mistakes include:

• Not following the order of operations
• Not isolating the variable on one side of the inequality
• Changing the direction of the inequality when dividing or multiplying both sides by a negative number

By avoiding these common mistakes and following the steps outlined in this article, you can confidently solve linear inequalities and determine which option represents all the solutions.

Real-World Applications

Linear inequalities have numerous real-world applications, including:

• Finance: Inequality $x \geq 1000$ can represent the minimum amount of money required to invest in a particular stock.
• Science: Inequality $x \leq 50$ can represent the maximum temperature at which a particular chemical reaction occurs.
• Engineering: Inequality $x \geq 20$ can represent the minimum strength required for a particular material to withstand a certain load.

Introduction

In our previous article, we solved the linear inequality $-5x + 5 \geq 160 - 10x$ and determined which of the given options represents all the solutions. In this article, we will provide a Q&A guide to help you better understand how to solve linear inequalities.

Q: What is a linear inequality?

A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form $ax + b$, where $a$ and $b$ are constants and $x$ is the variable.

A: How do I solve a linear inequality?

To solve a linear inequality, you need to isolate the variable on one side of the inequality. 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 order of operations when solving a linear inequality?

The order of operations when solving a linear inequality is the same as when solving an equation:

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: How do I handle negative numbers when solving a linear inequality?

When solving a linear inequality, you need to be careful when dividing or multiplying both sides of the inequality by a negative number. This can change the direction of the inequality. For example, if you have the inequality $x \geq 5$ and you divide both sides by $-2$, the inequality becomes $x \leq -\frac{5}{2}$.

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

A linear inequality involves a linear expression, while a quadratic inequality involves a quadratic expression, which is an expression that can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.

Q: How do I graph a linear inequality?

To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality. For example, if you have the inequality $x \geq 2$, you would graph the line $x = 2$ and then shade the region to the right of the line.

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

Some common mistakes to avoid when solving linear inequalities include:

• Not following the order of operations
• Not isolating the variable on one side of the inequality
• Changing the direction of the inequality when dividing or multiplying both sides by a negative number
• Not considering the restrictions on the variable

Q: How do I apply linear inequalities to real-world problems?

Linear inequalities have numerous real-world applications, including:

• Finance: Inequality $x \geq 1000$ can represent the minimum amount of money required to invest in a particular stock.
• Science: Inequality $x \leq 50$ can represent the maximum temperature at which a particular chemical reaction occurs.
• Engineering: Inequality $x \geq 20$ can represent the minimum strength required for a particular material to withstand a certain load.

By understanding how to solve linear inequalities, you can apply this knowledge to a wide range of real-world problems and make informed decisions.

Conclusion

In this article, we provided a Q&A guide to help you better understand how to solve linear inequalities. We covered topics such as the order of operations, handling negative numbers, and graphing linear inequalities. We also discussed common mistakes to avoid and provided examples of real-world applications of linear inequalities. By following the steps outlined in this article, you can confidently solve linear inequalities and apply this knowledge to a wide range of problems.