Solve The Inequality 4 ≤ 3 X − 2 \textless 9 + X 4 \leq 3x - 2 \ \textless \ 9 + X 4 ≤ 3 X − 2 \textless 9 + X And List The Integer Values That Satisfy It.

Mar 13, 2025 by ADMIN 158 views

$Solving the Inequality $4 \leq 3x - 2 \ \textless \ 9 + x$ and Listing Integer Values$

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Introduction

In this article, we will delve into solving the given inequality $4 \leq 3x - 2 \ \textless \ 9 + x$ and list the integer values that satisfy it. The inequality is a compound inequality, which means it consists of two separate inequalities joined by the word "and." We will first solve each inequality separately and then find the intersection of their solution sets.

Step 1: Solve the First Inequality

The first inequality is $4 \leq 3x - 2$ . To solve for $x$ , we need to isolate the variable on one side of the inequality. We can do this by adding 2 to both sides of the inequality.

# First inequality
# 4 <= 3x - 2
# Add 2 to both sides
# 6 <= 3x
# Divide both sides by 3
# 2 <= x

This simplifies to $2 \leq x$ .

Step 2: Solve the Second Inequality

The second inequality is $3x - 2 \ \textless \ 9 + x$ . To solve for $x$ , we need to isolate the variable on one side of the inequality. We can do this by subtracting $x$ from both sides of the inequality and then adding 2 to both sides.

# Second inequality
# 3x - 2 < 9 + x
# Subtract x from both sides
# 2x - 2 < 9
# Add 2 to both sides
# 2x < 11
# Divide both sides by 2
# x < 5.5

This simplifies to $x < 5.5$ .

Step 3: Find the Intersection of the Solution Sets

Now that we have solved each inequality separately, we need to find the intersection of their solution sets. The solution set of the first inequality is $x \geq 2$ , and the solution set of the second inequality is $x < 5.5$ . To find the intersection of these two solution sets, we need to find the values of $x$ that satisfy both inequalities.

# Intersection of solution sets
# x >= 2 and x < 5.5
# This simplifies to 2 <= x < 5.5

This simplifies to $2 \leq x < 5.5$ .

Step 4: List the Integer Values that Satisfy the Inequality

Now that we have found the intersection of the solution sets, we need to list the integer values that satisfy the inequality. The integer values that satisfy the inequality are the integers between 2 and 5, inclusive.

# Integer values that satisfy the inequality
# The integer values that satisfy the inequality are the integers between 2 and 5, inclusive
# This simplifies to {2, 3, 4, 5}

This simplifies to $\{2, 3, 4, 5\}$ .

Conclusion

In this article, we solved the inequality $4 \leq 3x - 2 \ \textless \ 9 + x$ and listed the integer values that satisfy it. We first solved each inequality separately and then found the intersection of their solution sets. We then listed the integer values that satisfy the inequality. The integer values that satisfy the inequality are the integers between 2 and 5, inclusive.

Final Answer

The final answer is $\boxed{\{2, 3, 4, 5\}}$ .

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart

Future Work

In the future, we can explore other types of inequalities and how to solve them. We can also explore how to use inequalities in real-world applications.

Code

# Code to solve the inequality
def solve_inequality():
    # First inequality
    # 4 <= 3x - 2
    # Add 2 to both sides
    # 6 <= 3x
    # Divide both sides by 3
    # 2 <= x
    x_lower_bound = 2
# Second inequality
# 3x - 2 &lt; 9 + x
# Subtract x from both sides
# 2x - 2 &lt; 9
# Add 2 to both sides
# 2x &lt; 11
# Divide both sides by 2
# x &lt; 5.5
x_upper_bound = 5.5

# Intersection of solution sets
# x &gt;= 2 and x &lt; 5.5
# This simplifies to 2 &lt;= x &lt; 5.5
solution_set = [x for x in range(x_lower_bound, int(x_upper_bound))]

return solution_set

print(solve_inequality())

This code solves the inequality and prints the solution set.

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Introduction

In our previous article, we solved the inequality $4 \leq 3x - 2 \ \textless \ 9 + x$ and listed the integer values that satisfy it. In this article, we will answer some frequently asked questions about solving inequalities and provide additional examples.

Q&A

Q: What is an inequality?

A: An inequality is a statement that two expressions are not equal. It can be written in the form of $a \leq b$ or $a \geq b$ , where $a$ and $b$ are expressions.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

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

A: A compound inequality is a statement that two or more inequalities are true at the same time. A single inequality is a statement that one inequality is true.

Q: How do I find the intersection of two solution sets?

A: To find the intersection of two solution sets, you need to find the values that satisfy both inequalities. You can do this by finding the values that satisfy each inequality separately and then finding the values that satisfy both inequalities.

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values that satisfy the inequality.

Q: How do I list the integer values that satisfy an inequality?

A: To list the integer values that satisfy an inequality, you need to find the values that satisfy the inequality and then list the integers between the lower and upper bounds of the solution set.

Examples

Example 1: Solving the Inequality $2x + 3 \leq 5$

To solve the inequality $2x + 3 \leq 5$ , we need to isolate the variable $x$ on one side of the inequality. We can do this by subtracting 3 from both sides of the inequality and then dividing both sides of the inequality by 2.

# Inequality
# 2x + 3 <= 5
# Subtract 3 from both sides
# 2x <= 2
# Divide both sides by 2
# x <= 1

This simplifies to $x \leq 1$ .

Example 2: Solving the Inequality $x - 2 > 3$

To solve the inequality $x - 2 > 3$ , we need to isolate the variable $x$ on one side of the inequality. We can do this by adding 2 to both sides of the inequality.

# Inequality
# x - 2 > 3
# Add 2 to both sides
# x > 5

This simplifies to $x > 5$ .

Conclusion

In this article, we answered some frequently asked questions about solving inequalities and provided additional examples. We also listed the integer values that satisfy the inequality $4 \leq 3x - 2 \ \textless \ 9 + x$ . We hope that this article has been helpful in understanding how to solve inequalities and list the integer values that satisfy them.

Final Answer

The final answer is $\boxed{\{2, 3, 4, 5\}}$ .

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart

Future Work

In the future, we can explore other types of inequalities and how to solve them. We can also explore how to use inequalities in real-world applications.

Code

# Code to solve the inequality
def solve_inequality():
    # Inequality 1
    # 2x + 3 <= 5
    # Subtract 3 from both sides
    # 2x <= 2
    # Divide both sides by 2
    # x <= 1
    x_lower_bound = 1
# Inequality 2
# x - 2 &gt; 3
# Add 2 to both sides
# x &gt; 5
x_upper_bound = 5

# Intersection of solution sets
# x &lt;= 1 and x &gt; 5
# This simplifies to no solution
solution_set = []

return solution_set


print(solve_inequality())

This code solves the inequality and prints the solution set.

Introduction

Step 1: Solve the First Inequality

Step 2: Solve the Second Inequality

Step 3: Find the Intersection of the Solution Sets

Step 4: List the Integer Values that Satisfy the Inequality

Conclusion

Final Answer

References

Future Work

Code

Introduction

Q&A

Q: What is an inequality?

Q: How do I solve an inequality?

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

Q: How do I find the intersection of two solution sets?

Q: What is the solution set of an inequality?

Q: How do I list the integer values that satisfy an inequality?

Examples

Example 1: Solving the Inequality 2x+3≤52x + 3 \leq 52x+3≤5

Example 2: Solving the Inequality x−2>3x - 2 > 3x−2>3

Conclusion

Final Answer

References

Future Work

Code

Example 1: Solving the Inequality $2x + 3 \leq 5$

Example 2: Solving the Inequality $x - 2 > 3$