Be Sure To Show All Work. Unsupported Answers May Receive No Credit.1. Find 3 Consecutive Odd Integers Whose Sum Is 165. (5 Points)2. Solve The Following Inequalities: 23 X − 3 − 7 ≤ 7 \frac{23x - 3}{-7} \leq 7 − 7 23 X − 3 ≤ 7 And − X \textless − ( 4 X − 9 -x \ \textless \ -(4x - 9 − X \textless − ( 4 X − 9 ].
Introduction
In this article, we will delve into two mathematical problems that require a deep understanding of algebra and problem-solving skills. The first problem involves finding three consecutive odd integers whose sum is 165, while the second problem requires solving two inequalities involving variables and constants. We will break down each problem step by step, providing a clear and concise explanation of the solution process.
Problem 1: Finding Consecutive Odd Integers
Problem Statement
Find three consecutive odd integers whose sum is 165.
Solution
Let's denote the first odd integer as x. Since we are looking for consecutive odd integers, the next two integers will be x + 2 and x + 4.
We know that the sum of these three integers is 165, so we can write the equation:
x + (x + 2) + (x + 4) = 165
Combine like terms:
3x + 6 = 165
Subtract 6 from both sides:
3x = 159
Divide both sides by 3:
x = 53
Now that we have found the value of x, we can find the next two consecutive odd integers:
x + 2 = 53 + 2 = 55 x + 4 = 53 + 4 = 57
Therefore, the three consecutive odd integers whose sum is 165 are 53, 55, and 57.
Problem 2: Solving Inequalities
Problem Statement
Solve the following inequalities:
Solution
Inequality 1
To solve the inequality , we need to isolate the variable x.
First, multiply both sides by -7:
23x - 3 ≥ -49
Add 3 to both sides:
23x ≥ -46
Divide both sides by 23:
x ≥ -46/23
x ≥ -2
Therefore, the solution to the inequality is x ≥ -2.
Inequality 2
To solve the inequality , we need to simplify the expression.
First, distribute the negative sign to the terms inside the parentheses:
-x < -4x + 9
Add 4x to both sides:
3x < 9
Divide both sides by 3:
x < 3
Therefore, the solution to the inequality is x < 3.
Conclusion
In this article, we have solved two mathematical problems that require a deep understanding of algebra and problem-solving skills. The first problem involved finding three consecutive odd integers whose sum is 165, while the second problem required solving two inequalities involving variables and constants. By breaking down each problem step by step, we have provided a clear and concise explanation of the solution process.
Key Takeaways
- When solving inequalities, it's essential to isolate the variable and simplify the expression.
- Consecutive odd integers can be found by starting with the first odd integer and adding 2 to find the next integer.
- Algebraic manipulations, such as multiplying and dividing by constants, can help simplify expressions and solve inequalities.
Final Thoughts
Introduction
In our previous article, we explored two mathematical problems that require a deep understanding of algebra and problem-solving skills. The first problem involved finding three consecutive odd integers whose sum is 165, while the second problem required solving two inequalities involving variables and constants. In this article, we will address some of the most frequently asked questions related to these problems.
Q&A
Q: What are consecutive odd integers?
A: Consecutive odd integers are a sequence of odd numbers that follow each other in order. For example, 1, 3, 5, 7, 9, etc.
Q: How do I find consecutive odd integers?
A: To find consecutive odd integers, start with the first odd integer and add 2 to find the next integer. For example, if the first odd integer is 1, the next odd integer would be 1 + 2 = 3, and the next one would be 3 + 2 = 5.
Q: What is the formula for finding consecutive odd integers?
A: The formula for finding consecutive odd integers is:
x, x + 2, x + 4, ...
where x is the first odd integer.
Q: How do I solve inequalities?
A: To solve inequalities, isolate the variable and simplify the expression. For example, if you have the inequality x + 2 > 5, subtract 2 from both sides to get x > 3.
Q: What is the difference between an inequality and an equation?
A: An inequality is a statement that compares two expressions using words like "greater than" or "less than." An equation is a statement that says two expressions are equal.
Q: How do I know which direction to multiply or divide when solving inequalities?
A: When solving inequalities, multiply or divide both sides by a positive number, and the direction of the inequality remains the same. When multiplying or dividing by a negative number, the direction of the inequality changes.
Q: Can I use the same steps to solve all types of inequalities?
A: No, not all inequalities can be solved using the same steps. Some inequalities may require additional steps or techniques, such as multiplying or dividing by a negative number.
Q: What are some common mistakes to avoid when solving inequalities?
A: Some common mistakes to avoid when solving inequalities include:
- Not isolating the variable
- Not simplifying the expression
- Not changing the direction of the inequality when multiplying or dividing by a negative number
Conclusion
In this article, we have addressed some of the most frequently asked questions related to consecutive odd integers and inequalities. By understanding these concepts and techniques, you can build confidence and develop problem-solving skills. Remember to always isolate the variable, simplify the expression, and change the direction of the inequality when necessary.
Key Takeaways
- Consecutive odd integers can be found by starting with the first odd integer and adding 2 to find the next integer.
- Inequalities can be solved by isolating the variable and simplifying the expression.
- When multiplying or dividing by a negative number, the direction of the inequality changes.
- Common mistakes to avoid when solving inequalities include not isolating the variable, not simplifying the expression, and not changing the direction of the inequality when necessary.
Final Thoughts
Solving mathematical problems requires patience, persistence, and a deep understanding of algebraic concepts. By addressing frequently asked questions and providing clear explanations, we can build confidence and develop problem-solving skills. Whether you're a student or a professional, mastering mathematical concepts is essential for success in various fields.