Solve $2x + 2 \ \textgreater \ 10$.A. $x \ \textgreater \ 6$ B. $ X \textgreater 4 X \ \textgreater \ 4 X \textgreater 4 [/tex] C. $x \ \textless \ 6$ D. $x \ \textless \ 4$

Mar 9, 2025 by ADMIN 195 views

**Solving Linear Inequalities: A Step-by-Step Guide**

Introduction

In mathematics, inequalities are used to compare two or more values. Linear inequalities are a type of inequality that involves a linear expression, such as a polynomial or a rational function. In this article, we will focus on solving linear inequalities of the form $ax + b > c$ , where $a$ , $b$ , and $c$ are constants.

What is a Linear Inequality?

A linear inequality is an inequality that involves a linear expression. It is a statement that one expression is greater than, less than, or equal to another expression. For example, $2x + 2 > 10$ is a linear inequality.

How to Solve a Linear Inequality

To solve a linear inequality, we need to isolate the variable on one side of the inequality sign. We can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Step 1: Simplify the Inequality

The first step in solving a linear inequality is to simplify the inequality by combining like terms. In the case of the inequality $2x + 2 > 10$ , we can simplify it by subtracting 2 from both sides:

2x + 2 - 2 > 10 - 2

This simplifies to:

2x > 8

Step 2: Isolate the Variable

The next step is to isolate the variable $x$ on one side of the inequality sign. We can do this by dividing both sides of the inequality by 2:

\frac{2x}{2} > \frac{8}{2}

This simplifies to:

x > 4

Step 3: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. In this case, the solution is $x > 4$ , which can be written as $(4, \infty)$ .

Solving the Given Inequality

Now that we have learned how to solve a linear inequality, let's apply this knowledge to the given inequality $2x + 2 > 10$ . We can follow the steps outlined above to solve this inequality:

Simplify the inequality by combining like terms:

2x + 2 - 2 > 10 - 2

This simplifies to:

2x > 8

Isolate the variable $x$ on one side of the inequality sign:

\frac{2x}{2} > \frac{8}{2}

This simplifies to:

x > 4

Therefore, the solution to the inequality $2x + 2 > 10$ is $x > 4$ .

Conclusion

In this article, we have learned how to solve linear inequalities of the form $ax + b > c$ . We have also applied this knowledge to the given inequality $2x + 2 > 10$ and found that the solution is $x > 4$ . We hope that this article has provided a clear and concise guide to solving linear inequalities.

Answer Key

The correct answer is:

A. $x > 6$
B. $x > 4$
C. $x < 6$
D. $x < 4$

Introduction

In our previous article, we discussed how to solve linear inequalities of the form $ax + b > c$ . In this article, we will provide a Q&A guide to help you better understand the concept of solving linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression. It is a statement that one expression is greater than, less than, or equal to another expression.

Q: How do I simplify a linear inequality?

A: To simplify a linear inequality, you need to combine like terms. This involves adding or subtracting the same value from both sides of the inequality.

Q: How do I isolate the variable in a linear inequality?

A: To isolate the variable, you need to divide both sides of the inequality by the same value. This will help you to get the variable on one side of the inequality sign.

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

A: A linear inequality involves a linear expression, while a quadratic inequality involves a quadratic expression. A linear inequality can be solved using basic algebraic operations, while a quadratic inequality requires more advanced techniques, such as factoring or the quadratic formula.

Q: Can I use the same steps to solve a linear inequality with a negative coefficient?

A: Yes, you can use the same steps to solve a linear inequality with a negative coefficient. 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 write the solution to a linear inequality in interval notation?

A: To write the solution to a linear inequality in interval notation, you need to use the following format:

If the inequality is of the form $x > a$ , the solution is $(a, \infty)$ .
If the inequality is of the form $x < a$ , the solution is $(-\infty, a)$ .
If the inequality is of the form $x = a$ , the solution is $\{a\}$ .

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

A: Yes, you can use a calculator to solve a linear inequality. However, you need to be careful when using a calculator, as it may not always give you the correct solution.

Q: How do I check my solution to a linear inequality?

A: To check your solution to a linear inequality, you need to plug in a value from the solution set into the original inequality. If the inequality is true, then your solution is correct.

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

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

Not simplifying the inequality before solving it.
Not isolating the variable correctly.
Not checking the solution.
Not using the correct interval notation.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concept of solving linear inequalities. We hope that this article has been helpful in answering your questions and providing you with a better understanding of this important mathematical concept.

Answer Key

The answers to the questions in this article are:

Q1: A linear inequality is an inequality that involves a linear expression.
Q2: To simplify a linear inequality, you need to combine like terms.
Q3: To isolate the variable, you need to divide both sides of the inequality by the same value.
Q4: A linear inequality involves a linear expression, while a quadratic inequality involves a quadratic expression.
Q5: Yes, you can use the same steps to solve a linear inequality with a negative coefficient.
Q6: To write the solution to a linear inequality in interval notation, you need to use the following format: $(a, \infty)$ , $(-\infty, a)$ , or $\{a\}$ .
Q7: Yes, you can use a calculator to solve a linear inequality.
Q8: To check your solution to a linear inequality, you need to plug in a value from the solution set into the original inequality.
Q9: Some common mistakes to avoid when solving linear inequalities include not simplifying the inequality, not isolating the variable correctly, not checking the solution, and not using the correct interval notation.