Solve \[$-2x - 7 \ \textgreater \ 5x + 14\$\].A. \[$x \ \textless \ -3\$\] B. \[$x \ \textgreater \ -3\$\] C. \[$x \ \textless \ -5\$\] D. \[$x \ \textgreater \ -5\$\]

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving the linear inequality ${-2x - 7 > 5x + 14}$. We will break down the solution process into manageable steps, making it easy for readers to understand and follow along.

Understanding the Inequality

Before we dive into solving the inequality, let's take a closer look at what it means. The inequality ${-2x - 7 > 5x + 14}$ states that the expression ${-2x - 7}$ is greater than the expression ${5x + 14}$. In other words, we are looking for values of ${x}$ that make the left-hand side of the inequality greater than the right-hand side.

Step 1: Add 7 to Both Sides

To solve the inequality, we need to isolate the variable ${x}$. The first step is to add 7 to both sides of the inequality. This will help us get rid of the constant term on the left-hand side.

${-2x - 7 + 7 > 5x + 14 + 7}$

Simplifying the inequality, we get:

${-2x > 5x + 21}$

Step 2: Subtract 5x from Both Sides

Next, we need to get rid of the term ${5x}$ on the right-hand side. We can do this by subtracting ${5x}$ from both sides of the inequality.

${-2x - 5x > 5x + 21 - 5x}$

Simplifying the inequality, we get:

${-7x > 21}$

Step 3: Divide Both Sides by -7

Now, we need to isolate the variable ${x}$. To do this, we can divide both sides of the inequality by ${-7}$. However, we need to be careful when dividing by a negative number. When we divide by a negative number, the direction of the inequality sign changes.

${\frac{-7x}{-7} > \frac{21}{-7}}$

Simplifying the inequality, we get:

${x < -3}$

Conclusion

In conclusion, the solution to the linear inequality ${-2x - 7 > 5x + 14}$ is ${x < -3}$. This means that any value of ${x}$ that is less than ${-3}$ will make the left-hand side of the inequality greater than the right-hand side.

Answer

The correct answer is:

• A. ${x < -3}$

Why is this the correct answer?

This is the correct answer because when we substitute ${x = -3}$ into the original inequality, we get:

${-2(-3) - 7 > 5(-3) + 14}$

Simplifying the inequality, we get:

${6 - 7 > -15 + 14}$

${-1 > -1}$

This is a false statement, which means that ${x = -3}$ does not satisfy the inequality. Therefore, the correct answer is ${x < -3}$.

Tips and Tricks

When solving linear inequalities, it's essential to follow the order of operations (PEMDAS):

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

Additionally, when dividing by a negative number, remember to change the direction of the inequality sign.

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving linear inequalities:

1. Solve the inequality ${3x + 2 > 5x - 3}$.
2. Solve the inequality ${2x - 4 < x + 2}$.
3. Solve the inequality ${x - 2 > 3x + 1}$.

Introduction

In our previous article, we discussed how to solve linear inequalities. In this article, we will provide a Q&A section to help clarify any doubts and provide additional examples to reinforce your understanding.

Q: What is a linear 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, and ${x}$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear 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 isolate the variable.
3. Change the direction of the inequality sign if you multiply or divide both sides by a negative number.

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

A: A linear equation is an equation that can be written in the form ${ax + b = c}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form ${ax + b > c}$ or ${ax + b < c}$.

Q: How do I know which direction to change the inequality sign when dividing by a negative number?

A: When dividing by a negative number, you change the direction of the inequality sign. For example, if you have the inequality ${x > 2}$ and you divide both sides by ${-3}$, the inequality becomes ${x < -\frac{2}{3}}$.

Q: Can I use the same steps to solve a quadratic inequality as I would a linear inequality?

A: No, you cannot use the same steps to solve a quadratic inequality as you would a linear inequality. Quadratic inequalities are more complex and require different techniques to solve.

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

A: A quadratic inequality is an inequality that can be written in the form ${ax^2 + bx + c > d}$ or ${ax^2 + bx + c < d}$, where ${a}$, ${b}$, ${c}$, and ${d}$ are constants, and ${x}$ is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form ${ax + b > c}$ or ${ax + b < c}$.

Q: How do I know if a solution to a linear inequality is an interval or a single value?

A: If the inequality is of the form ${x > a}$ or ${x < a}$, the solution is an interval. If the inequality is of the form ${x = a}$, the solution is a single value.

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

A: Yes, you can use a calculator to solve a linear inequality. However, make sure to check your work and verify that the solution is correct.

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving linear inequalities:

1. Solve the inequality ${2x + 3 > 5x - 2}$.
2. Solve the inequality ${x - 4 < 2x + 1}$.
3. Solve the inequality ${x^2 + 2x + 1 > 0}$.

I hope this Q&A article has helped clarify any doubts and provided additional examples to reinforce your understanding of solving linear inequalities. Remember to practice regularly to become proficient in solving these types of problems.