Solve The Following Inequality Using The Algebraic Approach:$\[ 5x - 1 \ \textless \ 2x + 11 \\]a. $\[ X \ \textless \ \frac{12}{7} \\] B. $\[ X \ \textless \ \frac{10}{3} \\] C. $\[ X \ \textless \ \frac{10}{7}

Introduction

In this article, we will focus on solving linear inequalities using the algebraic approach. Linear inequalities are mathematical expressions that contain a variable and a constant, and are often used to describe relationships between quantities. Solving linear inequalities involves isolating the variable on one side of the inequality sign, and can be a crucial skill in mathematics and real-world applications.

What are Linear Inequalities?

A linear inequality is a mathematical expression that contains a variable and a constant, and is often written in the form of:

ax + b < c

or

ax + b > c

where a, b, and c are constants, and x is the variable.

The Algebraic Approach

The algebraic approach to solving linear inequalities involves using the properties of inequalities to isolate the variable on one side of the inequality sign. This can be done by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Step 1: Isolate the Variable

To solve a linear inequality, we need to isolate the variable on one side of the inequality sign. This can be done by adding or subtracting the same value to both sides of the inequality.

Example 1: Solving the Inequality 5x - 1 < 2x + 11

To solve this inequality, we need to isolate the variable x on one side of the inequality sign.

${ 5x - 1 \ \textless \ 2x + 11 }$

First, we add 1 to both sides of the inequality:

${ 5x - 1 + 1 \ \textless \ 2x + 11 + 1 }$

This simplifies to:

${ 5x \ \textless \ 2x + 12 }$

Next, we subtract 2x from both sides of the inequality:

${ 5x - 2x \ \textless \ 2x + 12 - 2x }$

This simplifies to:

${ 3x \ \textless \ 12 }$

Finally, we divide both sides of the inequality by 3:

${ \frac{3x}{3} \ \textless \ \frac{12}{3} }$

This simplifies to:

${ x \ \textless \ \frac{12}{3} }$

Example 2: Solving the Inequality x - 2 < 3

To solve this inequality, we need to isolate the variable x on one side of the inequality sign.

${ x - 2 \ \textless \ 3 }$

First, we add 2 to both sides of the inequality:

${ x - 2 + 2 \ \textless \ 3 + 2 }$

This simplifies to:

${ x \ \textless \ 5 }$

Example 3: Solving the Inequality 2x + 1 > 3

To solve this inequality, we need to isolate the variable x on one side of the inequality sign.

${ 2x + 1 \ \textgreater \ 3 }$

First, we subtract 1 from both sides of the inequality:

${ 2x + 1 - 1 \ \textgreater \ 3 - 1 }$

This simplifies to:

${ 2x \ \textgreater \ 2 }$

Next, we divide both sides of the inequality by 2:

${ \frac{2x}{2} \ \textgreater \ \frac{2}{2} }$

This simplifies to:

${ x \ \textgreater \ 1 }$

Conclusion

Solving linear inequalities using the algebraic approach involves isolating the variable on one side of the inequality sign. This can be done by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value. By following the steps outlined in this article, you can solve linear inequalities and gain a deeper understanding of mathematical concepts.

Common Mistakes to Avoid

When solving linear inequalities, it's essential to avoid common mistakes that can lead to incorrect solutions. Some common mistakes to avoid include:

• Not isolating the variable: Make sure to isolate the variable on one side of the inequality sign.
• Not checking the direction of the inequality: Be careful when multiplying or dividing both sides of the inequality by a negative value, as this can change the direction of the inequality.
• Not considering the domain of the variable: Make sure to consider the domain of the variable when solving linear inequalities.

Real-World Applications

Linear inequalities have numerous real-world applications in fields such as economics, finance, and engineering. Some examples of real-world applications of linear inequalities include:

• Budgeting: Linear inequalities can be used to create budgets and track expenses.
• Investment: Linear inequalities can be used to determine the best investment options based on risk and return.
• Design: Linear inequalities can be used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

Q: What is a linear inequality?

A: A linear inequality is a mathematical expression that contains a variable and a constant, and is often written in the form of:

ax + b < c

or

ax + b > c

where a, b, and c are constants, and x is the variable.

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. This can be done by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What are the steps to solve a linear inequality?

A: The steps to solve a linear inequality are:

1. Isolate the variable on one side of the inequality sign.
2. Add or subtract the same value to both sides of the inequality.
3. Multiply or divide both sides of the inequality by the same non-zero value.
4. Check the direction of the inequality.

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

A: A linear equation is a mathematical expression that contains a variable and a constant, and is often written in the form of:

ax + b = c

where a, b, and c are constants, and x is the variable.

A linear inequality, on the other hand, is a mathematical expression that contains a variable and a constant, and is often written in the form of:

ax + b < c

or

ax + b > c

where a, b, and c are constants, and x is the variable.

Q: How do I know which direction to use when solving a linear inequality?

A: When solving a linear inequality, you need to determine the direction of the inequality. This can be done by checking the sign of the coefficient of the variable.

• If the coefficient of the variable is positive, use the "greater than" or "less than" symbol.
• If the coefficient of the variable is negative, use the "less than" or "greater than" symbol.

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

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

• Not isolating the variable on one side of the inequality sign.
• Not checking the direction of the inequality.
• Not considering the domain of the variable.

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

A: Linear inequalities have numerous real-world applications in fields such as economics, finance, and engineering. Some examples of real-world applications of linear inequalities include:

• Budgeting: Linear inequalities can be used to create budgets and track expenses.
• Investment: Linear inequalities can be used to determine the best investment options based on risk and return.
• Design: Linear inequalities can be used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: What are some tips for solving linear inequalities?

A: Some tips for solving linear inequalities include:

• Read the problem carefully and understand what is being asked.
• Use the correct steps to solve the inequality.
• Check the direction of the inequality.
• Consider the domain of the variable.

Conclusion

In conclusion, solving linear inequalities using the algebraic approach involves isolating the variable on one side of the inequality sign. By following the steps outlined in this article, you can solve linear inequalities and gain a deeper understanding of mathematical concepts. Remember to avoid common mistakes and consider the domain of the variable when solving linear inequalities.