What Is The Graph Of $-24 \leq 10x - 4 \ \textless \ 16$?

Introduction to Inequalities and Graphing

Inequalities are mathematical expressions that compare two values or expressions using greater than, less than, greater than or equal to, or less than or equal to. Graphing inequalities is a crucial concept in mathematics, particularly in algebra and geometry. It involves representing the solution set of an inequality on a number line or coordinate plane. In this article, we will explore the graph of the inequality $-24 \leq 10x - 4 \ \textless \ 16$.

Understanding the Inequality

The given inequality is a compound inequality, which means it consists of two separate inequalities combined by the word "and." The first part of the inequality is $-24 \leq 10x - 4$, and the second part is $10x - 4 \ \textless \ 16$. To graph this inequality, we need to solve each part separately and then find the intersection of the two solution sets.

Solving the First Inequality

To solve the first inequality, $-24 \leq 10x - 4$, we need to isolate the variable $x$. We can do this by adding 4 to both sides of the inequality, which gives us $-20 \leq 10x$. Next, we divide both sides by 10, resulting in $-2 \leq x$. This means that the solution set for the first inequality is all values of $x$ that are greater than or equal to -2.

Solving the Second Inequality

To solve the second inequality, $10x - 4 \ \textless \ 16$, we need to isolate the variable $x$ again. We can do this by adding 4 to both sides of the inequality, which gives us $10x \ \textless \ 20$. Next, we divide both sides by 10, resulting in $x \ \textless \ 2$. This means that the solution set for the second inequality is all values of $x$ that are less than 2.

Finding the Intersection of the Two Solution Sets

Now that we have solved both inequalities, we need to find the intersection of the two solution sets. The intersection of two sets is the set of elements that are common to both sets. In this case, the intersection of the two solution sets is the set of values of $x$ that satisfy both inequalities. Since the first inequality has a solution set of $x \geq -2$ and the second inequality has a solution set of $x \ \textless \ 2$, the intersection of the two solution sets is the set of values of $x$ that are greater than or equal to -2 and less than 2.

Graphing the Inequality

To graph the inequality, we can use a number line or a coordinate plane. We will use a number line to represent the solution set of the inequality. We start by marking the point -2 on the number line, which represents the lower bound of the solution set. We then draw an open circle at the point -2 to indicate that it is not included in the solution set. Next, we draw a closed circle at the point 2, which represents the upper bound of the solution set. We then draw a line segment connecting the two points, which represents the solution set of the inequality.

Conclusion

In conclusion, the graph of the inequality $-24 \leq 10x - 4 \ \textless \ 16$ is a line segment on a number line that represents the solution set of the inequality. The solution set is the set of values of $x$ that satisfy both inequalities, which is the set of values of $x$ that are greater than or equal to -2 and less than 2. We can use a number line or a coordinate plane to graph the inequality and represent the solution set.

Frequently Asked Questions

• What is the solution set of the inequality $-24 \leq 10x - 4 \ \textless \ 16$?
• How do I graph the inequality $-24 \leq 10x - 4 \ \textless \ 16$?
• What is the intersection of the two solution sets of the inequality $-24 \leq 10x - 4 \ \textless \ 16$?

Final Thoughts

Graphing inequalities is a crucial concept in mathematics, particularly in algebra and geometry. It involves representing the solution set of an inequality on a number line or coordinate plane. In this article, we explored the graph of the inequality $-24 \leq 10x - 4 \ \textless \ 16$. We solved each part of the inequality separately and then found the intersection of the two solution sets. We also graphed the inequality using a number line and represented the solution set.

Introduction

Graphing inequalities is a crucial concept in mathematics, particularly in algebra and geometry. It involves representing the solution set of an inequality on a number line or coordinate plane. In this article, we will answer some frequently asked questions about graphing inequalities.

Q&A

Q: What is the solution set of the inequality $-24 \leq 10x - 4 \ \textless \ 16$?

A: The solution set of the inequality $-24 \leq 10x - 4 \ \textless \ 16$ is the set of values of $x$ that satisfy both inequalities. To find the solution set, we need to solve each part of the inequality separately and then find the intersection of the two solution sets. The solution set is the set of values of $x$ that are greater than or equal to -2 and less than 2.

Q: How do I graph the inequality $-24 \leq 10x - 4 \ \textless \ 16$?

A: To graph the inequality $-24 \leq 10x - 4 \ \textless \ 16$, we can use a number line or a coordinate plane. We start by marking the point -2 on the number line, which represents the lower bound of the solution set. We then draw an open circle at the point -2 to indicate that it is not included in the solution set. Next, we draw a closed circle at the point 2, which represents the upper bound of the solution set. We then draw a line segment connecting the two points, which represents the solution set of the inequality.

Q: What is the intersection of the two solution sets of the inequality $-24 \leq 10x - 4 \ \textless \ 16$?

A: The intersection of the two solution sets of the inequality $-24 \leq 10x - 4 \ \textless \ 16$ is the set of values of $x$ that satisfy both inequalities. To find the intersection, we need to solve each part of the inequality separately and then find the common values of $x$ that satisfy both inequalities. The intersection of the two solution sets is the set of values of $x$ that are greater than or equal to -2 and less than 2.

Q: How do I determine the direction of the inequality symbol when graphing?

A: When graphing an inequality, the direction of the inequality symbol depends on the sign of the coefficient of the variable. If the coefficient is positive, the inequality symbol points to the right. If the coefficient is negative, the inequality symbol points to the left.

Q: Can I graph an inequality on a coordinate plane?

A: Yes, you can graph an inequality on a coordinate plane. To graph an inequality on a coordinate plane, you need to identify the boundary line and the direction of the inequality symbol. You can then shade the region on one side of the boundary line to represent the solution set of the inequality.

Q: How do I graph a compound inequality?

A: To graph a compound inequality, you need to graph each part of the inequality separately and then find the intersection of the two solution sets. You can use a number line or a coordinate plane to graph the inequality.

Q: Can I graph an inequality with a fraction?

A: Yes, you can graph an inequality with a fraction. To graph an inequality with a fraction, you need to simplify the fraction and then graph the inequality as you would any other inequality.

Conclusion

Graphing inequalities is a crucial concept in mathematics, particularly in algebra and geometry. It involves representing the solution set of an inequality on a number line or coordinate plane. In this article, we answered some frequently asked questions about graphing inequalities. We hope that this article has provided you with a better understanding of graphing inequalities and how to represent the solution set of an inequality on a number line or coordinate plane.

Final Thoughts

Graphing inequalities is a fundamental concept in mathematics, and it is essential to understand how to graph inequalities to solve problems in algebra and geometry. By following the steps outlined in this article, you can graph inequalities with confidence and represent the solution set of an inequality on a number line or coordinate plane.