Solve And Graph The Following Inequality: $\[ \frac{x}{3} - 4 \ \textless \ -2 \\]

Feb 28, 2025 by ADMIN 85 views

$Solve and Graph the Inequality: $\frac{x}{3} - 4 < -2$$

Introduction

In this article, we will focus on solving and graphing the given inequality $\frac{x}{3} - 4 < -2$ . This involves isolating the variable $x$ and understanding the relationship between the solution set and the graph of the inequality.

Understanding the Inequality

The given inequality is $\frac{x}{3} - 4 < -2$ . To begin solving this inequality, we need to isolate the variable $x$ . We can do this by adding $4$ to both sides of the inequality, which gives us $\frac{x}{3} < 2$ . This step is essential in understanding the relationship between the solution set and the graph of the inequality.

Isolating the Variable

To isolate the variable $x$ , we need to get rid of the fraction $\frac{1}{3}$ . We can do this by multiplying both sides of the inequality by $3$ , which gives us $x < 6$ . This step is crucial in understanding the solution set of the inequality.

Graphing the Inequality

To graph the inequality $x < 6$ , we need to draw a line at $x = 6$ and shade the region to the left of the line. This represents all the values of $x$ that satisfy the inequality.

Solution Set

The solution set of the inequality $x < 6$ is all real numbers less than $6$ . This can be represented as $(-\infty, 6)$ .

Conclusion

In conclusion, we have solved and graphed the inequality $\frac{x}{3} - 4 < -2$ . We have isolated the variable $x$ and understood the relationship between the solution set and the graph of the inequality. The solution set of the inequality is all real numbers less than $6$ , which can be represented as $(-\infty, 6)$ .

Step-by-Step Solution

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

Add $4$ to both sides of the inequality: $\frac{x}{3} - 4 + 4 < -2 + 4$
Simplify the inequality: $\frac{x}{3} < 2$
Multiply both sides of the inequality by $3$ : $3 \times \frac{x}{3} < 3 \times 2$
Simplify the inequality: $x < 6$

Graphing the Solution Set

To graph the solution set of the inequality $x < 6$ , we need to draw a line at $x = 6$ and shade the region to the left of the line. This represents all the values of $x$ that satisfy the inequality.

Real-World Applications

The inequality $x < 6$ has many real-world applications. For example, if we are planning a party and we want to make sure that everyone arrives before $6$ pm, we can use this inequality to determine the number of guests that can arrive on time.

Common Mistakes

When solving and graphing inequalities, there are many common mistakes that students make. Some of these mistakes include:

Not isolating the variable
Not understanding the relationship between the solution set and the graph of the inequality
Not shading the correct region on the graph

Tips and Tricks

Here are some tips and tricks for solving and graphing inequalities:

Always isolate the variable
Understand the relationship between the solution set and the graph of the inequality
Shade the correct region on the graph
Use a number line to help visualize the solution set

Conclusion

In conclusion, solving and graphing inequalities is an essential skill in mathematics. By following the steps outlined in this article, you can solve and graph any inequality. Remember to always isolate the variable, understand the relationship between the solution set and the graph of the inequality, and shade the correct region on the graph. With practice and patience, you will become proficient in solving and graphing inequalities.

Introduction

Solving and graphing inequalities can be a challenging task, especially for students who are new to mathematics. In this article, we will answer some of the most frequently asked questions about solving and graphing inequalities.

Q: What is an inequality?

A: An inequality is a statement that compares two expressions using a mathematical symbol such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing the same operations on both sides of the inequality.

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

A: A linear inequality is an inequality that can be written in the form ax + b < c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c < d, where a, b, c, and d are constants.

Q: How do I graph an inequality?

A: To graph an inequality, you need to draw a line at the value of x that makes the inequality true and shade the region on one side of the line.

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values of x that make the inequality true.

Q: How do I determine the solution set of an inequality?

A: To determine the solution set of an inequality, you need to isolate the variable and then determine the values of x that make the inequality true.

Q: What is the relationship between the solution set and the graph of an inequality?

A: The solution set of an inequality is the set of all values of x that make the inequality true, and the graph of an inequality is a visual representation of the solution set.

Q: How do I use a number line to help visualize the solution set of an inequality?

A: To use a number line to help visualize the solution set of an inequality, you need to mark the value of x that makes the inequality true and then shade the region on one side of the line.

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

A: Some common mistakes to avoid when solving and graphing inequalities include not isolating the variable, not understanding the relationship between the solution set and the graph of the inequality, and not shading the correct region on the graph.

Q: How can I practice solving and graphing inequalities?

A: You can practice solving and graphing inequalities by working through examples and exercises in a textbook or online resource, or by using a graphing calculator to visualize the solution set of an inequality.

Q: What are some real-world applications of solving and graphing inequalities?

A: Solving and graphing inequalities has many real-world applications, including determining the number of guests that can arrive on time for a party, calculating the cost of a product based on its weight, and determining the maximum height of a building based on its foundation.

Q: How can I improve my skills in solving and graphing inequalities?

A: You can improve your skills in solving and graphing inequalities by practicing regularly, seeking help from a teacher or tutor, and using online resources to visualize the solution set of an inequality.

Conclusion

Solving and graphing inequalities is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article and practicing regularly, you can improve your skills in solving and graphing inequalities. Remember to always isolate the variable, understand the relationship between the solution set and the graph of the inequality, and shade the correct region on the graph.