Which List Shows Three Solutions To The Inequality X + 34 ≥ 50 X + 34 \geq 50 X + 34 ≥ 50 ?A. 10, 15, 20B. 12, 16, 24C. 9, 11, 13D. 16, 20, 24

Mar 11, 2025 by ADMIN 143 views

**Solving Inequalities: A Step-by-Step Guide**

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more values. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $x + 34 \geq 50$ and provide three solutions to this inequality.

Understanding the Inequality

The given inequality is $x + 34 \geq 50$ . To solve this inequality, we need to isolate the variable $x$ . We can do this by subtracting 34 from both sides of the inequality.

Step 1: Subtract 34 from Both Sides

Subtracting 34 from both sides of the inequality gives us:

$x + 34 - 34 \geq 50 - 34$

Simplifying the inequality, we get:

$x \geq 16$

Step 2: Understanding the Solution

The solution to the inequality $x \geq 16$ means that the value of $x$ can be any number greater than or equal to 16. In other words, $x$ can be 16, 17, 18, 19, and so on.

Three Solutions to the Inequality

Now that we have the solution to the inequality, let's find three solutions that satisfy the inequality $x \geq 16$ . We can do this by finding three numbers that are greater than or equal to 16.

Solution 1: 16

The first solution to the inequality is 16. This is because 16 is greater than or equal to 16, which satisfies the inequality.

Solution 2: 20

The second solution to the inequality is 20. This is because 20 is greater than or equal to 16, which satisfies the inequality.

Solution 3: 24

The third solution to the inequality is 24. This is because 24 is greater than or equal to 16, which satisfies the inequality.

Conclusion

In conclusion, the three solutions to the inequality $x + 34 \geq 50$ are 16, 20, and 24. These solutions satisfy the inequality and provide a clear understanding of the concept of solving inequalities.

Final Answer

The final answer to the problem is:

A. 16, 20, 24

This is because the three solutions to the inequality $x + 34 \geq 50$ are 16, 20, and 24, which are listed in option A.

Additional Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

Always isolate the variable on one side of the inequality.
Use inverse operations to solve the inequality.
Check your solution by plugging it back into the original inequality.
Be careful when dealing with negative numbers and fractions.

By following these tips and tricks, you can become proficient in solving inequalities and tackle more complex problems with confidence.

Common Mistakes to Avoid

When solving inequalities, it's easy to make mistakes. Here are some common mistakes to avoid:

Not isolating the variable on one side of the inequality.
Not using inverse operations to solve the inequality.
Not checking the solution by plugging it back into the original inequality.
Not being careful when dealing with negative numbers and fractions.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Real-World Applications

Solving inequalities has numerous real-world applications. Here are a few examples:

In finance, inequalities are used to calculate interest rates and investment returns.
In engineering, inequalities are used to design and optimize systems.
In medicine, inequalities are used to model and analyze medical data.

By understanding and applying inequalities, you can make informed decisions and solve real-world problems with confidence.

Conclusion

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two or more values using a symbol such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality using inverse operations. This involves adding or subtracting the same value to both sides of the inequality, or multiplying or dividing both sides by the same non-zero value.

Q: What is the difference between an inequality and an equation?

A: An equation is a mathematical statement that states that two or more values are equal. An inequality, on the other hand, states that two or more values are not equal, but are related in some way.

Q: How do I know which direction to move the inequality symbol when solving an inequality?

A: When solving an inequality, you need to move the inequality symbol in the opposite direction of the operation you are performing. For example, if you are adding 3 to both sides of the inequality, you need to move the inequality symbol to the left.

Q: Can I multiply or divide both sides of an inequality by a negative number?

A: No, you cannot multiply or divide both sides of an inequality by a negative number. This is because it would change the direction of the inequality symbol, which would be incorrect.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to plug it back into the original inequality and see if it is true. If it is true, then your solution is correct.

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 on one side of the inequality
Not using inverse operations to solve the inequality
Not checking the solution by plugging it back into the original inequality
Not being careful when dealing with negative numbers and fractions

Q: How do I apply inequalities in real-world situations?

A: Inequalities are used in a variety of real-world situations, including:

Finance: Inequalities are used to calculate interest rates and investment returns.
Engineering: Inequalities are used to design and optimize systems.
Medicine: Inequalities are used to model and analyze medical data.

Q: Can I use inequalities to solve problems with multiple variables?

A: Yes, you can use inequalities to solve problems with multiple variables. This involves using systems of inequalities to represent the relationships between the variables.

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

A: To graph an inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality. You then draw an arrow on the number line to indicate the direction of the inequality.

Q: Can I use inequalities to solve problems with absolute values?

A: Yes, you can use inequalities to solve problems with absolute values. This involves using the definition of absolute value to rewrite the inequality in a form that can be solved using standard inequality techniques.

Conclusion

In conclusion, solving inequalities is a fundamental concept in mathematics that deals with the comparison of two or more values. By understanding and applying inequalities, you can solve a wide range of problems and make informed decisions in real-world situations. Remember to always isolate the variable, use inverse operations, and check your solution by plugging it back into the original inequality. By following these tips and tricks, you can become proficient in solving inequalities and tackle more complex problems with confidence.