Solve The Inequality: $\frac{x}{5} - \frac{2x - 1}{5} \geq 12$

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Introduction

Inequalities are mathematical expressions that compare two values, often with a greater than or less than sign. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality $\frac{x}{5} - \frac{2x - 1}{5} \geq 12$. We will break down the solution into manageable steps, making it easy to understand and follow.

Step 1: Simplify the Left-Hand Side

The first step in solving the inequality is to simplify the left-hand side by combining the fractions.

$\frac{x}{5} - \frac{2x - 1}{5} \geq 12$

To simplify, we can multiply both fractions by 5 to eliminate the denominator.

$x - (2x - 1) \geq 60$

Now, we can simplify the expression by combining like terms.

$-x + 1 \geq 60$

Step 2: Isolate the Variable

The next step is to isolate the variable $x$ on one side of the inequality sign. To do this, we can add $x$ to both sides of the inequality.

$-x + x + 1 \geq 60 + x$

This simplifies to:

$1 \geq 60 + x$

Step 3: Solve for $x$

Now that we have isolated the variable, we can solve for $x$. To do this, we can subtract 60 from both sides of the inequality.

$1 - 60 \geq x$

This simplifies to:

$-59 \geq x$

Step 4: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. Interval notation is a way of writing the solution to an inequality using parentheses and brackets.

$(-\infty, -59]$

This means that the solution to the inequality is all real numbers less than or equal to -59.

Conclusion

Solving inequalities involves breaking down the problem into manageable steps and following a logical process. In this article, we solved the inequality $\frac{x}{5} - \frac{2x - 1}{5} \geq 12$ by simplifying the left-hand side, isolating the variable, and solving for $x$. The final solution is $(-\infty, -59]$, which means that the solution to the inequality is all real numbers less than or equal to -59.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes to avoid. These include:

• Not simplifying the left-hand side: Failing to simplify the left-hand side can make it difficult to isolate the variable and solve for $x$.
• Not isolating the variable: Failing to isolate the variable can make it difficult to solve for $x$.
• Not checking the solution: Failing to check the solution can lead to incorrect answers.

Tips and Tricks

When solving inequalities, there are several tips and tricks to keep in mind. These include:

• Simplifying the left-hand side: Simplifying the left-hand side can make it easier to isolate the variable and solve for $x$.
• Using inverse operations: Using inverse operations can help to isolate the variable and solve for $x$.
• Checking the solution: Checking the solution can help to ensure that the answer is correct.

Real-World Applications

Solving inequalities has several real-world applications. These include:

• Finance: In finance, inequalities are used to model financial situations, such as investments and loans.
• Science: In science, inequalities are used to model physical situations, such as motion and energy.
• Engineering: In engineering, inequalities are used to model complex systems, such as electrical circuits and mechanical systems.

Conclusion

Introduction

In our previous article, we solved the inequality $\frac{x}{5} - \frac{2x - 1}{5} \geq 12$ using a step-by-step approach. In this article, we will answer some common questions related to solving inequalities.

Q: What is the difference between an inequality and an equation?

A: An inequality is a mathematical expression that compares two values, often with a greater than or less than sign. An equation, on the other hand, is a mathematical expression that states that two values are equal.

Q: How do I simplify the left-hand side of an inequality?

A: To simplify the left-hand side of an inequality, you can combine like terms and eliminate any common factors. For example, in the inequality $\frac{x}{5} - \frac{2x - 1}{5} \geq 12$, you can multiply both fractions by 5 to eliminate the denominator.

Q: How do I isolate the variable in an inequality?

A: To isolate the variable in an inequality, you can use inverse operations to get the variable on one side of the inequality sign. For example, in the inequality $-x + 1 \geq 60$, you can add $x$ to both sides to get $1 \geq 60 + x$.

Q: What is interval notation, and how do I use it to write the solution to an inequality?

A: Interval notation is a way of writing the solution to an inequality using parentheses and brackets. For example, the solution to the inequality $-59 \geq x$ is written as $(-\infty, -59]$.

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

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

• Not simplifying the left-hand side
• Not isolating the variable
• Not checking the solution

Q: What are some tips and tricks for solving inequalities?

A: Some tips and tricks for solving inequalities include:

• Simplifying the left-hand side
• Using inverse operations
• Checking the solution

Q: How do I check the solution to an inequality?

A: To check the solution to an inequality, you can plug in a value from the solution set into the original inequality and check if it is true.

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

A: Some real-world applications of solving inequalities include:

• Finance: In finance, inequalities are used to model financial situations, such as investments and loans.
• Science: In science, inequalities are used to model physical situations, such as motion and energy.
• Engineering: In engineering, inequalities are used to model complex systems, such as electrical circuits and mechanical systems.

Conclusion

Solving inequalities is an important skill in mathematics and has several real-world applications. By following a logical process and avoiding common mistakes, you can solve inequalities with ease. Remember to simplify the left-hand side, isolate the variable, and check the solution to ensure that the answer is correct.

Additional Resources

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

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

Practice Problems

Try solving the following inequalities:

• $\frac{x}{3} - \frac{2x + 1}{3} \geq 10$
• $-2x + 1 \geq 50$
• $x - 3 \geq 20$

Answer Key

• $\frac{x}{3} - \frac{2x + 1}{3} \geq 10$: $(-\infty, 31]$
• $-2x + 1 \geq 50$: $(-\infty, -24.5]$
• $x - 3 \geq 20$: $[23, \infty)$