Solve 2 X − 16 ≥ 32 − 6 X 2x - 16 \geq 32 - 6x 2 X − 16 ≥ 32 − 6 X .A. X ≥ − 12 X \geq -12 X ≥ − 12 B. X ≥ 6 X \geq 6 X ≥ 6 C. X ≤ − 12 X \leq -12 X ≤ − 12 D. X ≤ ∅ X \leq \emptyset X ≤ ∅

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Introduction

In this article, we will be solving the inequality $2x - 16 \geq 32 - 6x$. This is a linear inequality, which means it can be solved using basic algebraic operations. We will use the properties of inequalities to isolate the variable $x$ and find the solution set.

Understanding the Inequality

The given inequality is $2x - 16 \geq 32 - 6x$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. We can start by adding $6x$ to both sides of the inequality to get:

$$2x + 6x - 16 \geq 32$$

Combining Like Terms

Next, we can combine the like terms on the left-hand side of the inequality:

$$8x - 16 \geq 32$$

Adding 16 to Both Sides

Now, we can add 16 to both sides of the inequality to get:

$$8x \geq 48$$

Dividing Both Sides by 8

Finally, we can divide both sides of the inequality by 8 to get:

$$x \geq 6$$

Conclusion

Therefore, the solution to the inequality $2x - 16 \geq 32 - 6x$ is $x \geq 6$. This means that any value of $x$ that is greater than or equal to 6 is a solution to the inequality.

Checking the Answer Choices

Let's check the answer choices to see which one matches our solution:

• A. $x \geq -12$: This is not the correct solution, as our solution is $x \geq 6$, not $x \geq -12$.
• B. $x \geq 6$: This is the correct solution, as our solution is $x \geq 6$.
• C. $x \leq -12$: This is not the correct solution, as our solution is $x \geq 6$, not $x \leq -12$.
• D. $x \leq \emptyset$: This is not the correct solution, as our solution is $x \geq 6$, not $x \leq \emptyset$.

Final Answer

The final answer is B. $x \geq 6$.

Step-by-Step Solution

Here is the step-by-step solution to the inequality:

1. Add $6x$ to both sides of the inequality: $2x + 6x - 16 \geq 32$
2. Combine like terms: $8x - 16 \geq 32$
3. Add 16 to both sides of the inequality: $8x \geq 48$
4. Divide both sides of the inequality by 8: $x \geq 6$

Tips and Tricks

Here are some tips and tricks for solving linear inequalities:

• Always add or subtract the same value to both sides of the inequality.
• Combine like terms on both sides of the inequality.
• Use the properties of inequalities to isolate the variable.
• Check the answer choices to see which one matches your solution.

Practice Problems

Here are some practice problems for solving linear inequalities:

• Solve the inequality $3x + 2 \geq 5x - 1$.
• Solve the inequality $2x - 3 \leq x + 2$.
• Solve the inequality $x - 2 \geq 3x + 1$.

Conclusion

In this article, we solved the inequality $2x - 16 \geq 32 - 6x$ using basic algebraic operations. We used the properties of inequalities to isolate the variable $x$ and find the solution set. We also checked the answer choices to see which one matched our solution. We hope this article has been helpful in understanding how to solve linear inequalities.

Introduction

In our previous article, we solved the inequality $2x - 16 \geq 32 - 6x$ using basic algebraic operations. In this article, we will answer some frequently asked questions about solving linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \geq cx + d$ or $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ 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 $x$ on one side of the inequality sign. You can do this by adding or subtracting the same value to both sides of the inequality, and then combining like terms.

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

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

• Adding or subtracting the wrong value to both sides of the inequality.
• Not combining like terms on both sides of the inequality.
• Not checking the answer choices to see which one matches your solution.

Q: How do I check my answer when solving a linear inequality?

A: To check your answer when solving a linear inequality, you need to plug your solution back into the original inequality and see if it is true. If it is true, then your solution is correct.

Q: What are some tips and tricks for solving linear inequalities?

A: Some tips and tricks for solving linear inequalities include:

• Always add or subtract the same value to both sides of the inequality.
• Combine like terms on both sides of the inequality.
• Use the properties of inequalities to isolate the variable.
• Check the answer choices to see which one matches your solution.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, you need to make sure that you are using the correct calculator function and that you are entering the correct values.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the related linear equation and then shade the region that satisfies the inequality.

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

A: Some real-world applications of linear inequalities include:

• Budgeting: You can use linear inequalities to determine how much money you can spend on different items.
• Time management: You can use linear inequalities to determine how much time you have available for different activities.
• Science: You can use linear inequalities to model real-world phenomena, such as the growth of a population or the spread of a disease.

Q: Can I use linear inequalities to solve systems of equations?

A: Yes, you can use linear inequalities to solve systems of equations. However, you need to make sure that the inequalities are consistent and that the solution set is non-empty.

Q: How do I determine if a linear inequality is consistent or inconsistent?

A: To determine if a linear inequality is consistent or inconsistent, you need to check if the solution set is non-empty. If the solution set is non-empty, then the inequality is consistent. If the solution set is empty, then the inequality is inconsistent.

Q: What are some common types of linear inequalities?

A: Some common types of linear inequalities include:

• Linear inequalities with one variable: These are inequalities that can be written in the form $ax + b \geq cx + d$ or $ax + b \leq cx + d$.
• Linear inequalities with two variables: These are inequalities that can be written in the form $ax + by \geq cx + dy$ or $ax + by \leq cx + dy$.
• Linear inequalities with three variables: These are inequalities that can be written in the form $ax + by + cz \geq dx + ey + fz$ or $ax + by + cz \leq dx + ey + fz$.

Q: Can I use linear inequalities to solve quadratic equations?

A: Yes, you can use linear inequalities to solve quadratic equations. However, you need to make sure that the quadratic equation can be factored and that the solution set is non-empty.

Q: How do I determine if a quadratic equation can be factored?

A: To determine if a quadratic equation can be factored, you need to check if the equation can be written in the form $(x - r)(x - s) = 0$, where $r$ and $s$ are the roots of the equation.

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

A: Some common mistakes to avoid when solving quadratic equations using linear inequalities include:

• Not factoring the quadratic equation correctly.
• Not checking the solution set to see if it is non-empty.
• Not using the correct linear inequality to solve the quadratic equation.

Q: Can I use linear inequalities to solve polynomial equations?

A: Yes, you can use linear inequalities to solve polynomial equations. However, you need to make sure that the polynomial equation can be factored and that the solution set is non-empty.

Q: How do I determine if a polynomial equation can be factored?

A: To determine if a polynomial equation can be factored, you need to check if the equation can be written in the form $(x - r)(x - s)(x - t) = 0$, where $r$, $s$, and $t$ are the roots of the equation.

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

A: Some common mistakes to avoid when solving polynomial equations using linear inequalities include:

• Not factoring the polynomial equation correctly.
• Not checking the solution set to see if it is non-empty.
• Not using the correct linear inequality to solve the polynomial equation.

Conclusion

In this article, we answered some frequently asked questions about solving linear inequalities. We hope this article has been helpful in understanding how to solve linear inequalities and how to use them to solve real-world problems.