Solve And Graph Each Inequality: { -2x + 6 \ \textgreater \ 3x - 34$}$

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving and graphing them is a crucial skill for students to master. In this article, we will focus on solving and graphing the linear inequality ${-2x + 6 \ \textgreater \ 3x - 34}$. We will break down the steps involved in solving the inequality, and then graph the solution on a number line.

Understanding the Inequality

The given inequality is ${-2x + 6 \ \textgreater \ 3x - 34}$. To solve this inequality, we need to isolate the variable x on one side of the inequality sign. The inequality sign indicates that we are looking for values of x that make the expression on the left-hand side greater than the expression on the right-hand side.

Step 1: Add 2x to Both Sides

To isolate the variable x, we need to get rid of the term -2x on the left-hand side. We can do this by adding 2x to both sides of the inequality. This will give us:

${-2x + 6 + 2x \ \textgreater \ 3x - 34 + 2x}$

Simplifying the left-hand side, we get:

${6 \ \textgreater \ 5x - 34}$

Step 2: Add 34 to Both Sides

Next, we need to get rid of the constant term -34 on the right-hand side. We can do this by adding 34 to both sides of the inequality. This will give us:

${6 + 34 \ \textgreater \ 5x - 34 + 34}$

Simplifying the left-hand side, we get:

${40 \ \textgreater \ 5x}$

Step 3: Divide Both Sides by 5

Finally, we need to isolate the variable x by dividing both sides of the inequality by 5. This will give us:

${\frac{40}{5} \ \textgreater \ \frac{5x}{5}}$

Simplifying the left-hand side, we get:

${8 \ \textgreater \ x}$

Graphing the Solution

To graph the solution, we need to plot the number line and shade the region that satisfies the inequality. Since the inequality is x < 8, we will shade the region to the left of 8.

Conclusion

In this article, we solved and graphed the linear inequality ${-2x + 6 \ \textgreater \ 3x - 34}$. We broke down the steps involved in solving the inequality and then graphed the solution on a number line. By following these steps, students can master the skill of solving and graphing linear inequalities.

Tips and Tricks

• When solving linear inequalities, always isolate the variable on one side of the inequality sign.
• When graphing the solution, use a number line and shade the region that satisfies the inequality.
• Make sure to check your work by plugging in test values to ensure that the solution is correct.

Common Mistakes

• Failing to isolate the variable on one side of the inequality sign.
• Graphing the solution incorrectly by shading the wrong region.
• Not checking the work by plugging in test values.

Real-World Applications

Linear inequalities have many real-world applications, such as:

• Modeling population growth and decline
• Determining the maximum or minimum value of a function
• Finding the optimal solution to a problem

By mastering the skill of solving and graphing linear inequalities, students can apply these concepts to real-world problems and make informed decisions.

Practice Problems

• Solve and graph the linear inequality ${2x + 5 \ \textless \ 3x - 2}$.
• Solve and graph the linear inequality ${x - 3 \ \textgreater \ 2x + 1}$.
• Solve and graph the linear inequality ${4x - 2 \ \textless \ 2x + 5}$.

Introduction

In our previous article, we discussed how to solve and graph linear inequalities. However, we know that practice makes perfect, and there's no better way to reinforce your understanding than by answering questions and solving problems. In this article, we'll provide a Q&A section to help you better understand the concepts of solving and graphing linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form ax + b > c, ax + b < c, 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, 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, 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 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, ax + b < c, ax + b ≥ c, or ax + b ≤ c.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to plot the number line and shade the region that satisfies the inequality. If the inequality is in the form x > a, you will shade the region to the right of a. If the inequality is in the form x < a, you will shade the region to the left of a.

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

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

• Failing to isolate the variable on one side of the inequality sign
• Graphing the solution incorrectly by shading the wrong region
• Not checking the work by plugging in test values

Q: How do I check my work when solving and graphing linear inequalities?

A: To check your work, you can plug in test values into the inequality to ensure that the solution is correct. For example, if you are solving the inequality x > 2, you can plug in x = 3 and x = 1 to ensure that the solution is correct.

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

A: Linear inequalities have many real-world applications, such as:

• Modeling population growth and decline
• Determining the maximum or minimum value of a function
• Finding the optimal solution to a problem

Q: How can I practice solving and graphing linear inequalities?

A: You can practice solving and graphing linear inequalities by working through practice problems, such as those found in textbooks or online resources. You can also try solving and graphing linear inequalities on your own, using real-world examples or scenarios.

Practice Problems

• Solve and graph the linear inequality ${2x + 5 \ \textless \ 3x - 2}$.
• Solve and graph the linear inequality ${x - 3 \ \textgreater \ 2x + 1}$.
• Solve and graph the linear inequality ${4x - 2 \ \textless \ 2x + 5}$.

Additional Resources

• Khan Academy: Linear Inequalities
• Mathway: Linear Inequalities
• Wolfram Alpha: Linear Inequalities

By practicing and reviewing the concepts of solving and graphing linear inequalities, you can become more confident and proficient in solving these types of problems. Remember to check your work and use real-world examples to help you understand the concepts better.