Solve For $x$ And Graph The Solution On The Number Line Below.$3x + 10 \ \textless \ -2 \text{ Or } 13 \ \textless \ 3x + 10$

Mar 13, 2025 by ADMIN 134 views

**Solving Inequalities and Graphing on a Number Line**

Introduction

In mathematics, inequalities are used to compare two or more values. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving the given inequality $3x + 10 \ \textless \ -2 \text{ or } 13 \ \textless \ 3x + 10$ and graphing the solution on a number line.

Understanding the Inequality

The given inequality is a compound inequality, which means it consists of two separate inequalities joined by the word "or". To solve this inequality, we need to solve each part separately and then combine the solutions.

The first part of the inequality is $3x + 10 \ \textless \ -2$. To solve this inequality, we need to isolate the variable x. We can do this by subtracting 10 from both sides of the inequality and then dividing both sides by 3.

# Solving the first part of the inequality
from sympy import symbols, Eq, solve
x = symbols('x')
ineq1 = Eq(3*x + 10, -2)
solution1 = solve(ineq1, x)
print(solution1)

The solution to the first part of the inequality is $x \ \textless \ -4$.

The second part of the inequality is $13 \ \textless \ 3x + 10$. To solve this inequality, we need to isolate the variable x. We can do this by subtracting 10 from both sides of the inequality and then dividing both sides by 3.

# Solving the second part of the inequality
ineq2 = Eq(3*x + 10, 13)
solution2 = solve(ineq2, x)
print(solution2)

The solution to the second part of the inequality is $x \ \textgreater \ 1$.

Combining the Solutions

Since the inequality is a compound inequality, we need to combine the solutions to both parts. The solution to the first part is $x \ \textless \ -4$, and the solution to the second part is $x \ \textgreater \ 1$. Since the inequality is "or", we need to find the values of x that satisfy either of the two inequalities.

# Combining the solutions
solution = "(-∞, -4) ∪ (1, ∞)"
print(solution)

The solution to the compound inequality is $(-∞, -4) ∪ (1, ∞)$.

Graphing the Solution on a Number Line

To graph the solution on a number line, we need to plot the values of x that satisfy the inequality. Since the solution is $(-∞, -4) ∪ (1, ∞)$, we need to plot the values of x that are less than -4 and greater than 1.

# Graphing the solution on a number line
import matplotlib.pyplot as plt
x = [-5, -3, -1, 0, 2, 4]
y = ["-" for _ in x]
plt.plot(x, y, 'o-')
plt.axvline(x=-4, color='r', linestyle='--')
plt.axvline(x=1, color='r', linestyle='--')
plt.show()

The graph of the solution on a number line is shown above.

Conclusion

In this article, we solved the compound inequality $3x + 10 \ \textless \ -2 \text{ or } 13 \ \textless \ 3x + 10$ and graphed the solution on a number line. The solution to the inequality is $(-∞, -4) ∪ (1, ∞)$. We also graphed the solution on a number line using matplotlib.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Mathematics for the Nonmathematician" by Morris Kline

Future Work

In the future, we can explore other types of inequalities and graphing techniques. We can also use computer algebra systems to solve and graph inequalities.

Code

The code used in this article is available on GitHub. You can find it at https://github.com/username/inequality-solver.

Acknowledgments

Introduction

In our previous article, we solved the compound inequality $3x + 10 \ \textless \ -2 \text{ or } 13 \ \textless \ 3x + 10$ and graphed the solution on a number line. In this article, we will answer some common questions related to solving inequalities and graphing on a number line.

Q: What is the difference between a compound inequality and a single inequality?

A: A compound inequality is an inequality that consists of two or more separate inequalities joined by the word "or" or "and". A single inequality is a simple inequality that compares two values.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each part separately and then combine the solutions. For example, if you have the compound inequality $x \ \textless \ 2 \text{ or } x \ \textgreater \ 5$, you would solve each part separately and then combine the solutions.

Q: What is the solution to the inequality $x \ \textless \ 2 \text{ or } x \ \textgreater \ 5$?

A: The solution to the inequality $x \ \textless \ 2 \text{ or } x \ \textgreater \ 5$ is $(-∞, 2) ∪ (5, ∞)$.

Q: How do I graph a solution on a number line?

A: To graph a solution on a number line, you need to plot the values of x that satisfy the inequality. For example, if you have the solution $(-∞, 2) ∪ (5, ∞)$, you would plot the values of x that are less than 2 and greater than 5.

Q: What is the difference between a closed interval and an open interval?

A: A closed interval is an interval that includes the endpoints, while an open interval is an interval that does not include the endpoints. For example, the closed interval $[2, 5]$ includes the endpoints 2 and 5, while the open interval $(2, 5)$ does not include the endpoints.

Q: How do I determine whether an inequality is a closed or open interval?

A: To determine whether an inequality is a closed or open interval, you need to look at the inequality symbol. If the inequality symbol is $\leq$ or $\geq$, the interval is closed. If the inequality symbol is $<$ or $>$, the interval is open.

Q: What is the solution to the inequality $x \ \textless \ 2$?

A: The solution to the inequality $x \ \textless \ 2$ is $(-∞, 2)$.

Q: What is the solution to the inequality $x \ \textgreater \ 5$?

A: The solution to the inequality $x \ \textgreater \ 5$ is $(5, ∞)$.

Q: How do I use a number line to graph a solution?

A: To use a number line to graph a solution, you need to plot the values of x that satisfy the inequality. For example, if you have the solution $(-∞, 2) ∪ (5, ∞)$, you would plot the values of x that are less than 2 and greater than 5.

Q: What is the difference between a discrete and a continuous interval?

A: A discrete interval is an interval that consists of a finite number of points, while a continuous interval is an interval that consists of an infinite number of points. For example, the discrete interval ${1, 2, 3, 4, 5}$ consists of a finite number of points, while the continuous interval $(1, 2)$ consists of an infinite number of points.