Which Value Is A Solution Of The Inequality $9 \geq \frac{1}{2} X$?A. 18 B. 20 C. 24 D. 36

Feb 26, 2025 by ADMIN 98 views

Solving Inequalities: A Step-by-Step Guide

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Introduction

In mathematics, inequalities are a fundamental concept that helps us compare values and make decisions. Solving inequalities involves finding the values of variables that satisfy a given inequality. In this article, we will focus on solving the inequality $9 \geq \frac{1}{2} x$ and determine which value is a solution to this inequality.

Understanding the Inequality

The given inequality is $9 \geq \frac{1}{2} x$. To solve this inequality, we need to isolate the variable x. The first step is to multiply both sides of the inequality by 2 to eliminate the fraction.

Multiplying Both Sides of the Inequality

When we multiply both sides of the inequality by 2, we get:

18 \geq x

This means that the value of x must be less than or equal to 18.

Analyzing the Options

Now that we have the solution to the inequality, let's analyze the options given:

A. 18
B. 20
C. 24
D. 36

Option A: 18

Option A is 18, which is the upper bound of the solution to the inequality. Since 18 is less than or equal to 18, it satisfies the inequality.

Option B: 20

Option B is 20, which is greater than 18. Since 20 is not less than or equal to 18, it does not satisfy the inequality.

Option C: 24

Option C is 24, which is also greater than 18. Since 24 is not less than or equal to 18, it does not satisfy the inequality.

Option D: 36

Option D is 36, which is also greater than 18. Since 36 is not less than or equal to 18, it does not satisfy the inequality.

Conclusion

Based on our analysis, the only option that satisfies the inequality is:

A. 18

Therefore, the value that is a solution to the inequality $9 \geq \frac{1}{2} x$ is 18.

Frequently Asked Questions

Q: What is the solution to the inequality $9 \geq \frac{1}{2} x$?

A: The solution to the inequality is $x \leq 18$.

Q: Which option satisfies the inequality?

A: Option A, 18, satisfies the inequality.

Q: Why do options B, C, and D not satisfy the inequality?

A: Options B, C, and D are greater than 18, which means they do not satisfy the inequality.

Final Thoughts

Solving inequalities is an essential skill in mathematics that helps us make decisions and compare values. By following the steps outlined in this article, you can solve inequalities and determine which values satisfy a given inequality. Remember to always analyze the options carefully and check if they satisfy the inequality.

Additional Resources

For more information on solving inequalities, check out the following resources:

Khan Academy: Solving Inequalities
Mathway: Solving Inequalities
Wolfram Alpha: Solving Inequalities

By following these resources and practicing solving inequalities, you can become proficient in solving inequalities and make informed decisions in mathematics and real-life situations.

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Introduction

In our previous article, we discussed how to solve the inequality $9 \geq \frac{1}{2} x$ and determined that the value that is a solution to this inequality is 18. In this article, we will provide a Q&A guide to help you better understand solving inequalities.

Q&A Guide

Q: What is an inequality?

A: An inequality is a statement that compares two values using a mathematical symbol, such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤).

Q: How do I solve an inequality?

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

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form ax + b > c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c > 0, where a, b, and c are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can follow these steps:

Isolate the variable on one side of the inequality sign.
Add, subtract, multiply, or divide both sides of the inequality by the same value.
Simplify the inequality to get the solution.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can follow these steps:

Factor the quadratic expression on the left-hand side of the inequality.
Set each factor equal to zero and solve for the variable.
Use a number line or a graph to determine the solution to the inequality.

Q: What is the solution to the inequality $x - 3 > 2$?

A: To solve this inequality, we need to isolate the variable x. We can do this by adding 3 to both sides of the inequality:

x - 3 + 3 > 2 + 3

This simplifies to:

x > 5

Therefore, the solution to the inequality is x > 5.

Q: What is the solution to the inequality $x^2 + 4x + 4 \leq 0$?

A: To solve this inequality, we need to factor the quadratic expression on the left-hand side:

(x + 2)^2 \leq 0

Since the square of any real number is always non-negative, the only way for this inequality to be true is if the expression inside the square is equal to zero:

(x + 2)^2 = 0

This implies that x + 2 = 0, which means x = -2.

Therefore, the solution to the inequality is x ≤ -2.