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Mar 9, 2025 by ADMIN 361 views

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

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Introduction

Linear inequalities are a fundamental concept in algebra and mathematics. They are used to describe relationships between variables and constants, and are essential in solving real-world problems. In this article, we will guide you through the step-by-step process of solving linear inequalities, using the inequality ${-10 + 8x < 6x - 4}$ as an example.

Step 1: Simplify the Inequality

The first step in solving the inequality is to simplify it by combining like terms. We can start by subtracting ${6x}$ from both sides of the inequality:

${-10 + 8x < 6x - 4}$

Subtracting ${6x}$ from both sides gives us:

${-10 + 2x < -4}$

Next, we can add ${10}$ to both sides of the inequality to isolate the variable term:

${2x < 6}$

This is the simplified form of the inequality.

Step 2: Isolate the Variable

The next step is to isolate the variable ${x}$ by dividing both sides of the inequality by the coefficient of ${x}$ , which is ${2}$ :

${2x < 6}$

Dividing both sides by ${2}$ gives us:

${x < 3}$

This is the final step in isolating the variable.

Step 3: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. The solution ${x < 3}$ can be written as:

${(-\infty, 3)}$

This represents all values of ${x}$ that are less than ${3}$ .

Step 4: Check the Solution

To check the solution, we can substitute a value of ${x}$ that is less than ${3}$ into the original inequality. Let's choose ${x = 0}$ :

${-10 + 8(0) < 6(0) - 4}$

Simplifying the inequality gives us:

${-10 < -4}$

This is true, so the solution ${x < 3}$ is correct.

Conclusion

Solving linear inequalities requires a step-by-step approach. By simplifying the inequality, isolating the variable, and writing the solution in interval notation, we can find the final solution. In this article, we used the inequality ${-10 + 8x < 6x - 4}$ as an example and walked through the steps to solve it. We also checked the solution to ensure that it is correct.

Tips and Tricks

When solving linear inequalities, it's essential to follow the order of operations (PEMDAS) to simplify the inequality.
When isolating the variable, make sure to divide both sides of the inequality by the coefficient of the variable.
When writing the solution in interval notation, use parentheses to represent the solution set.

Common Mistakes

Failing to simplify the inequality before isolating the variable.
Dividing both sides of the inequality by a coefficient that is zero.
Writing the solution in interval notation without using parentheses.

Real-World Applications

Linear inequalities have numerous real-world applications, including:

Finance: Linear inequalities are used to model financial relationships, such as investment returns and interest rates.
Science: Linear inequalities are used to model scientific relationships, such as population growth and chemical reactions.
Engineering: Linear inequalities are used to model engineering relationships, such as stress and strain in materials.

Practice Problems

Try solving the following linear inequalities:

${2x + 5 < 3x - 2}$
${x - 3 > 2x + 1}$
${4x - 2 < 2x + 6}$

Conclusion

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Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form ${ax + b < c}$ or ${ax + b > c}$ , where ${a}$ , ${b}$ , and ${c}$ are constants, and ${x}$ is the variable.

Q: How do I simplify a linear inequality?

A: To simplify a linear inequality, you can combine like terms on both sides of the inequality. For example, if you have the inequality ${2x + 3 < 5x - 2}$ , you can combine the like terms on the left-hand side to get ${3 < 3x - 2}$ .

Q: How do I isolate the variable in a linear inequality?

A: To isolate the variable in a linear inequality, you can add or subtract the same value from both sides of the inequality. For example, if you have the inequality ${2x + 3 < 5x - 2}$ , you can subtract ${2x}$ from both sides to get ${3 < 3x - 2}$ . Then, you can add ${2}$ to both sides to get ${5 < 3x}$ .

Q: How do I write the solution in interval notation?

A: To write the solution in interval notation, you can use parentheses to represent the solution set. For example, if the solution is ${x < 3}$ , you can write it as ${(-\infty, 3)}$ .

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

A: A linear equation is an equation that can be written in the form ${ax + b = c}$ , where ${a}$ , ${b}$ , and ${c}$ are constants, and ${x}$ is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form ${ax + b < c}$ or ${ax + b > c}$ .

Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?

A: No, you cannot use the same steps to solve a linear inequality as you would to solve a linear equation. When solving a linear inequality, you need to follow the steps of simplifying the inequality, isolating the variable, and writing the solution in interval notation.

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

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

Failing to simplify the inequality before isolating the variable.
Dividing both sides of the inequality by a coefficient that is zero.
Writing the solution in interval notation without using parentheses.

Q: How do I check the solution to a linear inequality?

A: To check the solution to a linear inequality, you can substitute a value of ${x}$ that is in the solution set into the original inequality. If the inequality is true, then the solution is correct.

Q: Can I use a calculator to solve a linear inequality?

A: Yes, you can use a calculator to solve a linear inequality. However, you need to be careful when using a calculator to solve an inequality, as it may not always give you the correct solution.

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

A: Linear inequalities have numerous real-world applications, including:

Finance: Linear inequalities are used to model financial relationships, such as investment returns and interest rates.
Science: Linear inequalities are used to model scientific relationships, such as population growth and chemical reactions.
Engineering: Linear inequalities are used to model engineering relationships, such as stress and strain in materials.

Q: Can I use linear inequalities to solve systems of equations?

A: Yes, you can use linear inequalities to solve systems of equations. However, you need to be careful when using linear inequalities to solve a system of equations, as it may not always give you the correct solution.

Q: What are some tips for solving linear inequalities?

A: Some tips for solving linear inequalities include:

Simplifying the inequality before isolating the variable.
Using interval notation to represent the solution set.
Checking the solution to ensure that it is correct.

Q: Can I use linear inequalities to solve quadratic equations?

A: No, you cannot use linear inequalities to solve quadratic equations. However, you can use linear inequalities to solve systems of linear equations that involve quadratic equations.

Q: What are some common mistakes to avoid when using linear inequalities to solve systems of equations?

A: Some common mistakes to avoid when using linear inequalities to solve systems of equations include:

Failing to simplify the inequality before isolating the variable.
Dividing both sides of the inequality by a coefficient that is zero.
Writing the solution in interval notation without using parentheses.

Q: Can I use linear inequalities to solve optimization problems?

A: Yes, you can use linear inequalities to solve optimization problems. However, you need to be careful when using linear inequalities to solve an optimization problem, as it may not always give you the correct solution.

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

A: Linear inequalities have numerous real-world applications in optimization, including:

Finance: Linear inequalities are used to model financial relationships, such as investment returns and interest rates.
Science: Linear inequalities are used to model scientific relationships, such as population growth and chemical reactions.
Engineering: Linear inequalities are used to model engineering relationships, such as stress and strain in materials.

Q: Can I use linear inequalities to solve linear programming problems?

A: Yes, you can use linear inequalities to solve linear programming problems. However, you need to be careful when using linear inequalities to solve a linear programming problem, as it may not always give you the correct solution.

Q: What are some common mistakes to avoid when using linear inequalities to solve linear programming problems?

A: Some common mistakes to avoid when using linear inequalities to solve linear programming problems include:

Failing to simplify the inequality before isolating the variable.
Dividing both sides of the inequality by a coefficient that is zero.
Writing the solution in interval notation without using parentheses.