Solve The Compound Inequality:$\[ 3x + 2 \ \textless \ -7 \quad \text{or} \quad 4x + 3 \geq 23 \\]Graph The Solution On The Number Line.

Feb 28, 2025 by ADMIN 139 views

Solving Compound Inequalities: A Step-by-Step Guide

=====================================================

Introduction

Compound inequalities are a type of mathematical expression that involves two or more inequalities joined by the words "and" or "or." In this article, we will focus on solving compound inequalities of the form $a \leq x \leq b$ or $a \geq x \geq b$ , where $a$ and $b$ are constants. We will also explore how to graph the solution on a number line.

Understanding Compound Inequalities

A compound inequality is a statement that combines two or more inequalities using the words "and" or "or." For example, the compound inequality $x > 2$ or $x < 5$ can be read as "x is greater than 2 or x is less than 5." In this case, the solution to the compound inequality is the union of the solutions to the individual inequalities.

Types of Compound Inequalities

Inequalities Joined by "or"

When two inequalities are joined by the word "or," the solution is the union of the solutions to the individual inequalities. This means that the solution includes all values that satisfy either of the two inequalities.

Example 1: Solving a Compound Inequality Joined by "or"

Solve the compound inequality $3x + 2 < -7$ or $4x + 3 \geq 23$ .

To solve this compound inequality, we need to solve each individual inequality separately.

Step 1: Solve the First Inequality

The first inequality is $3x + 2 < -7$ . To solve for $x$ , we need to isolate the variable $x$ .

from sympy import symbols, Eq, solve

# Define the variable
x = symbols('x')

# Define the inequality
ineq = Eq(3*x + 2, -7)

# Solve the inequality
solution = solve(ineq, x)

print(solution)

The solution to the first inequality is $x < -3$ .

Step 2: Solve the Second Inequality

The second inequality is $4x + 3 \geq 23$ . To solve for $x$ , we need to isolate the variable $x$ .

from sympy import symbols, Eq, solve

# Define the variable
x = symbols('x')

# Define the inequality
ineq = Eq(4*x + 3, 23)

# Solve the inequality
solution = solve(ineq, x)

print(solution)

The solution to the second inequality is $x \geq 5$ .

Step 3: Find the Union of the Solutions

The solution to the compound inequality is the union of the solutions to the individual inequalities. In this case, the solution is $x < -3$ or $x \geq 5$ .

Graphing the Solution on a Number Line

To graph the solution on a number line, we need to plot the points that satisfy the inequality. In this case, the points that satisfy the inequality are $x = -3$ and $x = 5$ .

Step 1: Plot the Points

To plot the points, we need to mark the points on the number line.

Step 2: Shade the Region

To shade the region, we need to shade the area between the points.

Conclusion

In this article, we learned how to solve compound inequalities of the form $a \leq x \leq b$ or $a \geq x \geq b$ . We also learned how to graph the solution on a number line. By following the steps outlined in this article, you should be able to solve compound inequalities and graph the solution on a number line.

Frequently Asked Questions

Q: What is a compound inequality?

A: A compound inequality is a statement that combines two or more inequalities using the words "and" or "or."

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each individual inequality separately and then find the union of the solutions.

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

A: To graph the solution on a number line, you need to plot the points that satisfy the inequality and then shade the region between the points.

References

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

Additional Resources

[1] Khan Academy: Solving Inequalities
[2] Mathway: Solving Inequalities
[3] Wolfram Alpha: Solving Inequalities

=====================================================

Introduction

In our previous article, we explored the concept of compound inequalities and how to solve them. However, we know that math can be a complex and confusing subject, and sometimes it's helpful to have a Q&A guide to clarify any doubts. In this article, we will provide a Q&A guide to help you better understand compound inequalities and how to solve them.

Q&A Guide

Q: What is a compound inequality?

A: A compound inequality is a statement that combines two or more inequalities using the words "and" or "or."

Q: How do I know if an inequality is a compound inequality?

A: If an inequality has two or more parts joined by the words "and" or "or," it is a compound inequality.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each individual inequality separately and then find the union of the solutions.

Q: What is the union of the solutions?

A: The union of the solutions is the set of all values that satisfy either of the two inequalities.

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

A: To graph the solution on a number line, you need to plot the points that satisfy the inequality and then shade the region between the points.

Q: What if the compound inequality has more than two parts?

A: If the compound inequality has more than two parts, you need to solve each individual inequality separately and then find the intersection of the solutions.

Q: What is the intersection of the solutions?

A: The intersection of the solutions is the set of all values that satisfy all of the inequalities.

Q: How do I know if the intersection of the solutions is empty?

A: If the intersection of the solutions is empty, it means that there are no values that satisfy all of the inequalities.

Q: What if the compound inequality has a variable in the middle?

A: If the compound inequality has a variable in the middle, you need to isolate the variable and then solve the inequality.

Q: How do I know if the compound inequality is true or false?

A: To determine if the compound inequality is true or false, you need to substitute a value into the inequality and see if it is true or false.

Q: What if the compound inequality has a negative sign?

A: If the compound inequality has a negative sign, you need to flip the inequality sign when multiplying or dividing both sides by a negative number.

Q: How do I know if the compound inequality is an equality or an inequality?

A: To determine if the compound inequality is an equality or an inequality, you need to look at the inequality sign.

Q: What if the compound inequality has a fraction?

A: If the compound inequality has a fraction, you need to multiply both sides by the reciprocal of the fraction to eliminate the fraction.

Q: How do I know if the compound inequality is a linear or non-linear inequality?

A: To determine if the compound inequality is a linear or non-linear inequality, you need to look at the inequality sign and the coefficients of the variables.

Conclusion

In this Q&A guide, we have covered some of the most common questions and answers related to compound inequalities. We hope that this guide has helped you better understand compound inequalities and how to solve them.

Frequently Asked Questions

Q: What is a compound inequality?

A: A compound inequality is a statement that combines two or more inequalities using the words "and" or "or."

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each individual inequality separately and then find the union of the solutions.

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

A: To graph the solution on a number line, you need to plot the points that satisfy the inequality and then shade the region between the points.

References

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

Additional Resources

[1] Khan Academy: Solving Inequalities
[2] Mathway: Solving Inequalities
[3] Wolfram Alpha: Solving Inequalities