Linear inequalities with one variable

Oct 13, 2025 by pynkocean.com 143 views

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. One of the most common types of inequalities is the linear inequality, which involves a linear expression with one or more variables. In this article, we will focus on solving a specific type of linear inequality, which involves fractions and variables. We will use the given inequality as an example to demonstrate the step-by-step process of solving it.

The Given Inequality

The given inequality is:

\frac{3(x-2)}{2} - \frac{7(x-3)}{4} \geq 3

This inequality involves two fractions with variables in the numerator and constants in the denominator. Our goal is to isolate the variable x and find the solution set that satisfies the inequality.

Step 1: Simplify the Inequality

To simplify the inequality, we need to find a common denominator for the two fractions. The least common multiple (LCM) of 2 and 4 is 4. We can rewrite the inequality as:

\frac{3(x-2)}{2} \cdot \frac{2}{2} - \frac{7(x-3)}{4} \geq 3

\frac{6(x-2)}{4} - \frac{7(x-3)}{4} \geq 3

Now, we can combine the two fractions by subtracting the numerators:

\frac{6(x-2) - 7(x-3)}{4} \geq 3

Step 2: Expand and Simplify the Numerator

To expand and simplify the numerator, we need to distribute the coefficients to the terms inside the parentheses:

\frac{6x - 12 - 7x + 21}{4} \geq 3

Now, we can combine like terms:

\frac{-x + 9}{4} \geq 3

Step 3: Multiply Both Sides by 4

To eliminate the fraction, we can multiply both sides of the inequality by 4:

-x + 9 \geq 12

Step 4: Isolate the Variable x

To isolate the variable x, we need to subtract 9 from both sides of the inequality:

-x \geq 3

Now, we can multiply both sides of the inequality by -1 to get:

x \leq -3

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear inequality involving fractions and variables. We started by simplifying the inequality, expanding and simplifying the numerator, multiplying both sides by 4, and finally isolating the variable x. The solution to the inequality is x ≤ -3.

Tips and Tricks

When solving linear inequalities, it's essential to follow the order of operations (PEMDAS) and to simplify the inequality as much as possible. Additionally, when multiplying or dividing both sides of the inequality by a negative number, the direction of the inequality sign may change.

Real-World Applications

Linear inequalities have numerous real-world applications in various fields, such as economics, finance, and engineering. For example, in economics, linear inequalities can be used to model the relationship between variables such as supply and demand. In finance, linear inequalities can be used to calculate the minimum or maximum value of an investment. In engineering, linear inequalities can be used to design and optimize systems.

Common Mistakes to Avoid

When solving linear inequalities, it's essential to avoid common mistakes such as:

Not simplifying the inequality enough
Not following the order of operations (PEMDAS)
Not changing the direction of the inequality sign when multiplying or dividing both sides by a negative number

By following these tips and tricks, you can avoid common mistakes and become proficient in solving linear inequalities.

Practice Problems

To practice solving linear inequalities, try the following problems:

Solve the inequality: 2x + 5 ≥ 11
Solve the inequality: x - 3 ≤ 2
Solve the inequality: 4x - 2 ≥ 10

Conclusion

Introduction

In our previous article, we discussed the step-by-step process of solving linear inequalities involving fractions and variables. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression with one or more variables. It is a mathematical statement that compares two expressions and indicates whether one is greater than, less than, or equal to the other.

Q: What are the different types of linear inequalities?

A: There are two main types of linear inequalities:

Linear inequalities with one variable: These inequalities involve a single variable and a linear expression.
Linear inequalities with two variables: These inequalities involve two variables and a linear expression.

Q: How do I simplify a linear inequality?

A: To simplify a linear inequality, you need to follow these steps:

Combine like terms: Combine any like terms in the numerator and denominator.
Simplify the numerator: Simplify the numerator by combining like terms and factoring out any common factors.
Simplify the denominator: Simplify the denominator by combining like terms and factoring out any common factors.
Cancel out any common factors: Cancel out any common factors in the numerator and denominator.

Q: How do I multiply or divide both sides of a linear inequality by a negative number?

A: When multiplying or dividing both sides of a linear inequality by a negative number, you need to change the direction of the inequality sign. For example:

If the original inequality is x ≥ 3 and you multiply both sides by -1, the new inequality becomes x ≤ -3.
If the original inequality is x ≤ 3 and you divide both sides by -1, the new inequality becomes x ≥ -3.

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

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

Not simplifying the inequality enough
Not following the order of operations (PEMDAS)
Not changing the direction of the inequality sign when multiplying or dividing both sides by a negative number
Not checking the solution set for extraneous solutions

Q: How do I check the solution set for extraneous solutions?

A: To check the solution set for extraneous solutions, you need to plug the solution back into the original inequality and check if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

Q: What are some real-world applications of linear inequalities?

A: Linear inequalities have numerous real-world applications in various fields, such as:

Economics: Linear inequalities can be used to model the relationship between variables such as supply and demand.
Finance: Linear inequalities can be used to calculate the minimum or maximum value of an investment.
Engineering: Linear inequalities can be used to design and optimize systems.

Q: How can I practice solving linear inequalities?

A: You can practice solving linear inequalities by trying the following problems:

Solve the inequality: 2x + 5 ≥ 11
Solve the inequality: x - 3 ≤ 2
Solve the inequality: 4x - 2 ≥ 10

You can also try solving linear inequalities with two variables, such as:

Solve the inequality: x + y ≥ 3
Solve the inequality: x - y ≤ 2

Conclusion

In conclusion, solving linear inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the step-by-step process outlined in this article and practicing with examples, you can become proficient in solving linear inequalities and apply them to various fields. Remember to simplify the inequality, expand and simplify the numerator, multiply both sides by 4, and finally isolate the variable x. With practice and patience, you can master the art of solving linear inequalities.