Solve The Compound Inequality: ${3x + 5 \ \textless \ 23 \quad \text{or} \quad 2x + 1 \ \textgreater \ 5}$Write The Solution In Interval Notation. If There Is No Solution, Enter { \varnothing$}$.

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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<ba \lt b or c>d.c \gt d. We will use the given compound inequality ${3x + 5 \ \textless \ 23 \quad \text{or} \quad 2x + 1 \ \textgreater \ 5}$ as an example to demonstrate the steps involved in solving compound inequalities.

Understanding Compound Inequalities

Compound inequalities can be solved using the same techniques as single inequalities. However, when solving compound inequalities, we need to consider two separate inequalities and find the solution set for each inequality. The solution set for the compound inequality is the union of the solution sets for each individual inequality.

Step 1: Solve the First Inequality

The first inequality is 3x+5<23.3x + 5 \lt 23. To solve this inequality, we need to isolate the variable x.x. We can do this by subtracting 5 from both sides of the inequality and then dividing both sides by 3.

# Import necessary modules
import sympy as sp

x = sp.symbols('x')



inequality1 = 3*x + 5 < 23



solution1 = sp.solve(inequality1, x)


print(solution1)

Step 2: Solve the Second Inequality

The second inequality is 2x+1>5.2x + 1 \gt 5. To solve this inequality, we need to isolate the variable x.x. We can do this by subtracting 1 from both sides of the inequality and then dividing both sides by 2.

# Define the inequality
inequality2 = 2*x + 1 > 5


solution2 = sp.solve(inequality2, x)


print(solution2)

Step 3: Find the Solution Set for Each Inequality

The solution set for the first inequality is x<6.x \lt 6. The solution set for the second inequality is x>2.x \gt 2.

Step 4: Find the Union of the Solution Sets

The solution set for the compound inequality is the union of the solution sets for each individual inequality. In this case, the solution set is x<6βˆͺx>2.x \lt 6 \cup x \gt 2.

Step 5: Write the Solution in Interval Notation

The solution set x<6βˆͺx>2x \lt 6 \cup x \gt 2 can be written in interval notation as (βˆ’βˆž,2)βˆͺ(6,∞).(-\infty, 2) \cup (6, \infty).

Conclusion

Solving compound inequalities involves finding the solution set for each individual inequality and then finding the union of the solution sets. In this article, we used the given compound inequality ${3x + 5 \ \textless \ 23 \quad \text{or} \quad 2x + 1 \ \textgreater \ 5}$ as an example to demonstrate the steps involved in solving compound inequalities. We found the solution set for each individual inequality and then found the union of the solution sets to obtain the final solution.

Example Problems

Problem 1

Solve the compound inequality ${2x - 3 \lt 7 \quad \text{or} \quad x + 2 \gt 9}$ and write the solution in interval notation.

Solution

To solve this compound inequality, we need to follow the same steps as before. First, we solve the first inequality 2xβˆ’3<72x - 3 \lt 7 to obtain x<5.x \lt 5. Then, we solve the second inequality x+2>9x + 2 \gt 9 to obtain x>7.x \gt 7. Finally, we find the union of the solution sets to obtain the final solution x<5βˆͺx>7.x \lt 5 \cup x \gt 7.

Problem 2

Solve the compound inequality ${x - 2 \gt 3 \quad \text{or} \quad 2x + 1 \lt 11}$ and write the solution in interval notation.

Solution

To solve this compound inequality, we need to follow the same steps as before. First, we solve the first inequality xβˆ’2>3x - 2 \gt 3 to obtain x>5.x \gt 5. Then, we solve the second inequality 2x+1<112x + 1 \lt 11 to obtain x<5.x \lt 5. Finally, we find the union of the solution sets to obtain the final solution x>5βˆͺx<5.x \gt 5 \cup x \lt 5.

Final Answer

The final answer is (βˆ’βˆž,2)βˆͺ(6,∞).\boxed{(-\infty, 2) \cup (6, \infty)}.

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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 provide a Q&A guide to help you understand and solve compound inequalities.

Q&A

Q: What is a compound inequality?

A: A compound inequality is a type of mathematical expression that involves two or more inequalities joined by the words "and" or "or."

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to follow these steps:

  1. Solve each individual inequality separately.
  2. Find the solution set for each individual inequality.
  3. Find the union of the solution sets to obtain the final solution.

Q: What is the union of the solution sets?

A: The union of the solution sets is the set of all values that satisfy at least one of the individual inequalities.

Q: How do I write the solution in interval notation?

A: To write the solution in interval notation, you need to use the following notation:

  • (βˆ’βˆž,a)(-\infty, a) to represent all values less than aa
  • (a,∞)(a, \infty) to represent all values greater than aa
  • (βˆ’βˆž,a]βˆͺ[a,∞)(-\infty, a] \cup [a, \infty) to represent all values less than or equal to aa or greater than or equal to aa

Q: What is the difference between "and" and "or" in compound inequalities?

A: In compound inequalities, "and" is used to represent the intersection of the solution sets, while "or" is used to represent the union of the solution sets.

Q: How do I solve a compound inequality with "and"?

A: To solve a compound inequality with "and," you need to find the intersection of the solution sets. This means that the solution set must satisfy both inequalities.

Q: How do I solve a compound inequality with "or"?

A: To solve a compound inequality with "or," you need to find the union of the solution sets. This means that the solution set must satisfy at least one of the inequalities.

Q: What is the final answer for the compound inequality ${3x + 5 \ \textless \ 23 \quad \text{or} \quad 2x + 1 \ \textgreater \ 5}$?

A: The final answer is (βˆ’βˆž,2)βˆͺ(6,∞).\boxed{(-\infty, 2) \cup (6, \infty)}.

Q: How do I check my answer?

A: To check your answer, you need to plug in a value from the solution set into the original inequality and verify that it is true.

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

A: Some common mistakes to avoid when solving compound inequalities include:

  • Not solving each individual inequality separately
  • Not finding the solution set for each individual inequality
  • Not finding the union of the solution sets
  • Not writing the solution in interval notation

Conclusion

Solving compound inequalities can be challenging, but with practice and patience, you can master this skill. Remember to follow the steps outlined in this article, and don't hesitate to ask for help if you need it. Good luck!

Example Problems

Problem 1

Solve the compound inequality ${2x - 3 \lt 7 \quad \text{or} \quad x + 2 \gt 9}$ and write the solution in interval notation.

Solution

To solve this compound inequality, you need to follow the same steps as before. First, you solve the first inequality 2xβˆ’3<72x - 3 \lt 7 to obtain x<5.x \lt 5. Then, you solve the second inequality x+2>9x + 2 \gt 9 to obtain x>7.x \gt 7. Finally, you find the union of the solution sets to obtain the final solution x<5βˆͺx>7.x \lt 5 \cup x \gt 7.

Problem 2

Solve the compound inequality ${x - 2 \gt 3 \quad \text{or} \quad 2x + 1 \lt 11}$ and write the solution in interval notation.

Solution

To solve this compound inequality, you need to follow the same steps as before. First, you solve the first inequality xβˆ’2>3x - 2 \gt 3 to obtain x>5.x \gt 5. Then, you solve the second inequality 2x+1<112x + 1 \lt 11 to obtain x<5.x \lt 5. Finally, you find the union of the solution sets to obtain the final solution x>5βˆͺx<5.x \gt 5 \cup x \lt 5.

Final Answer

The final answer is (βˆ’βˆž,2)βˆͺ(6,∞).\boxed{(-\infty, 2) \cup (6, \infty)}.