First Natural Number that Satisfies the Inequality

Oct 13, 2025 by talynt.org 131 views

Introduction

Inequalities are mathematical expressions that compare two values, indicating whether one is greater than, less than, or equal to the other. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving a specific inequality: $2 - \frac{3x}{4} < 14$ . We will break down the solution process into manageable steps and provide a clear explanation of each step.

Step 1: Subtract 2 from Both Sides

To begin solving the inequality, we need to isolate the term containing the variable, $x$ . The first step is to subtract 2 from both sides of the inequality.

$2 - \frac{3x}{4} < 14$

Subtracting 2 from both sides gives us:

$-\frac{3x}{4} < 12$

Step 2: Multiply Both Sides by -4

To eliminate the fraction, we need to multiply both sides of the inequality by -4. However, when multiplying or dividing both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

$-\frac{3x}{4} < 12$

Multiplying both sides by -4 gives us:

$3x > -48$

Step 3: Divide Both Sides by 3

Now that we have isolated the variable, $x$ , we can divide both sides of the inequality by 3 to solve for $x$ .

$3x > -48$

Dividing both sides by 3 gives us:

$x > -16$

Conclusion

We have successfully solved the inequality $2 - \frac{3x}{4} < 14$ . The solution is $x > -16$ . This means that any value of $x$ greater than -16 will satisfy the inequality.

First Natural Number that Satisfies the Inequality

The first natural number that satisfies the inequality is the smallest natural number greater than -16. Natural numbers are positive integers, so the first natural number greater than -16 is 1.

Discussion

In this article, we have demonstrated how to solve a linear inequality. We have broken down the solution process into manageable steps and provided a clear explanation of each step. Solving inequalities is an essential skill in mathematics, and it has numerous applications in real-world problems.

Tips and Tricks

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

When multiplying or dividing both sides of an inequality by a negative number, reverse the direction of the inequality sign.
When adding or subtracting the same value from both sides of an inequality, the direction of the inequality sign remains the same.
When multiplying or dividing both sides of an inequality by a positive number, the direction of the inequality sign remains the same.

By following these tips and tricks, you can confidently solve linear inequalities and apply them to real-world problems.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid the following common mistakes:

Reversing the direction of the inequality sign when multiplying or dividing both sides by a negative number.
Failing to reverse the direction of the inequality sign when multiplying or dividing both sides by a negative number.
Not checking the direction of the inequality sign after adding or subtracting the same value from both sides.

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, including:

Finance: Inequality can be used to model financial situations, such as determining the minimum amount of money needed to save for a specific goal.
Science: Inequality can be used to model scientific situations, such as determining the minimum amount of time needed to complete a task.
Engineering: Inequality can be used to model engineering situations, such as determining the minimum amount of material needed to build a structure.

By applying inequality to real-world problems, you can develop a deeper understanding of mathematical concepts and their practical applications.

Conclusion

Introduction

In our previous article, we discussed how to solve linear inequalities. In this article, we will provide a Q&A guide to help you better understand the concept of solving inequalities. We will cover common questions and concerns that students often have when solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, indicating whether one is greater than, less than, or equal to the other.

Q: What are the different types of inequalities?

A: There are two main types of inequalities: linear inequalities and quadratic inequalities. Linear inequalities involve a single variable and a linear expression, while quadratic inequalities involve a single variable and a quadratic expression.

Q: How do I solve a linear inequality?

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

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

A: A linear inequality involves a single variable and a linear expression, while a quadratic inequality involves a single variable and a quadratic expression. Quadratic inequalities are more complex and require different solution techniques.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign chart method to determine the solution set.

Q: What is the sign chart method?

A: The sign chart method is a technique used to solve quadratic inequalities. You create a chart with the critical points of the quadratic expression and then determine the sign of the expression in each interval.

Q: How do I determine the critical points of a quadratic expression?

A: To determine the critical points of a quadratic expression, you need to find the values of the variable that make the expression equal to zero.

Q: What is the solution set of an inequality?

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

Q: How do I graph an inequality?

A: To graph an inequality, you need to graph the related equation and then shade the region that satisfies the inequality.

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

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

Reversing the direction of the inequality sign when multiplying or dividing both sides by a negative number.
Failing to reverse the direction of the inequality sign when multiplying or dividing both sides by a negative number.
Not checking the direction of the inequality sign after adding or subtracting the same value from both sides.

Q: How do I apply inequalities to real-world problems?

A: Inequalities have numerous real-world applications, including finance, science, and engineering. You can use inequalities to model financial situations, scientific phenomena, and engineering problems.

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can confidently solve linear and quadratic inequalities and apply them to real-world problems. Remember to avoid common mistakes and follow tips and tricks to ensure accurate and reliable solutions.

Additional Resources

For further learning, we recommend the following resources:

Khan Academy: Inequalities
Mathway: Inequalities
Wolfram Alpha: Inequalities

By practicing and applying inequalities to real-world problems, you can develop a deeper understanding of mathematical concepts and their practical applications.