Solve For $x$: X 5 − 1 ≤ 2 − X 10 \frac{x}{5}-1 \leq 2-\frac{x}{10} 5 X ​ − 1 ≤ 2 − 10 X ​

Introduction

In mathematics, solving inequalities is a crucial skill that helps us understand the relationship between different variables. In this article, we will focus on solving the inequality $\frac{x}{5}-1 \leq 2-\frac{x}{10}$, which involves fractions and variables. We will break down the solution step by step, using algebraic manipulations and logical reasoning to find the value of $x$ that satisfies the given inequality.

Understanding the Inequality

The given inequality is $\frac{x}{5}-1 \leq 2-\frac{x}{10}$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to simplify the inequality by combining like terms.

Simplifying the Inequality

To simplify the inequality, we can start by getting rid of the fractions. We can do this by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators, which is 10.

\frac{x}{5}-1 \leq 2-\frac{x}{10}
\Rightarrow 10\left(\frac{x}{5}-1\right) \leq 10\left(2-\frac{x}{10}\right)
\Rightarrow 2x-10 \leq 20-x

Isolating the Variable

Now that we have simplified the inequality, we can isolate the variable $x$ by moving all the terms involving $x$ to one side of the inequality sign.

2x-10 \leq 20-x
\Rightarrow 2x+x \leq 20+10
\Rightarrow 3x \leq 30
\Rightarrow x \leq 10

Checking the Solution

To check our solution, we can plug in a value of $x$ that satisfies the inequality and see if it holds true. Let's say we choose $x=5$. Plugging this value into the original inequality, we get:

\frac{5}{5}-1 \leq 2-\frac{5}{10}
\Rightarrow 1-1 \leq 2-\frac{1}{2}
\Rightarrow 0 \leq \frac{3}{2}

Since this is true, we can be confident that our solution is correct.

Conclusion

In this article, we solved the inequality $\frac{x}{5}-1 \leq 2-\frac{x}{10}$ by simplifying it, isolating the variable, and checking the solution. We found that the value of $x$ that satisfies the inequality is $x \leq 10$. This solution is important in mathematics, as it helps us understand the relationship between different variables and make predictions about real-world phenomena.

Applications of the Solution

The solution to this inequality has many applications in real-world scenarios. For example, in finance, it can be used to determine the maximum amount of money that can be invested in a particular stock. In engineering, it can be used to design systems that meet certain performance criteria. In science, it can be used to model the behavior of complex systems and make predictions about their behavior.

Tips for Solving Inequalities

Solving inequalities can be challenging, but with practice and patience, it can become a breeze. Here are some tips for solving inequalities:

• Start by simplifying the inequality by combining like terms.
• Use algebraic manipulations to isolate the variable.
• Check the solution by plugging in a value of the variable that satisfies the inequality.
• Use logical reasoning to determine the correct solution.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes to avoid. Here are some of the most common mistakes:

• Not simplifying the inequality before solving it.
• Not isolating the variable correctly.
• Not checking the solution.
• Not using logical reasoning to determine the correct solution.

Final Thoughts

Solving inequalities is an important skill that has many applications in real-world scenarios. By following the steps outlined in this article, you can solve inequalities with confidence and make predictions about complex systems. Remember to simplify the inequality, isolate the variable, and check the solution. With practice and patience, you can become a master of solving inequalities.

Introduction

In our previous article, we solved the inequality $\frac{x}{5}-1 \leq 2-\frac{x}{10}$ and found that the value of $x$ that satisfies the inequality is $x \leq 10$. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to simplify it by combining like terms. This involves getting rid of any fractions or decimals by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.

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

A: To isolate the variable in an inequality, you need to move all the terms involving the variable to one side of the inequality sign. This can be done by adding or subtracting the same value to both sides of the inequality.

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

A: A linear inequality is an inequality that can be written in the form $ax+b \leq cx+d$, where $a$, $b$, $c$, and $d$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2+bx+c \leq dx^2+ex+f$, where $a$, $b$, $c$, $d$, $e$, and $f$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to first factor the quadratic expression on the left-hand side of the inequality. Then, you need to find the values of $x$ that make the quadratic expression equal to zero. These values are called the roots of the quadratic equation. Finally, you need to test each interval between the roots to see which one satisfies the inequality.

Q: What is the importance of checking the solution in an inequality?

A: Checking the solution in an inequality is important because it ensures that the solution is correct. If you don't check the solution, you may end up with an incorrect answer.

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

A: To check the solution in an inequality, you need to plug in a value of $x$ that satisfies the inequality and see if it holds true. If the inequality holds true, then the solution is correct.

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

A: Some common mistakes to avoid when solving inequalities include not simplifying the inequality before solving it, not isolating the variable correctly, not checking the solution, and not using logical reasoning to determine the correct solution.

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through examples and exercises in a textbook or online resource. You can also try solving inequalities on your own and then checking your answers with a calculator or online tool.

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

A: Solving inequalities has many real-world applications, including finance, engineering, and science. For example, in finance, inequalities can be used to determine the maximum amount of money that can be invested in a particular stock. In engineering, inequalities can be used to design systems that meet certain performance criteria. In science, inequalities can be used to model the behavior of complex systems and make predictions about their behavior.

Q: How can I improve my skills in solving inequalities?

A: To improve your skills in solving inequalities, you need to practice regularly and consistently. You can also try working with a tutor or mentor who can provide you with guidance and support. Additionally, you can try using online resources and tools to help you solve inequalities.

Conclusion

Solving inequalities is an important skill that has many real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence and make predictions about complex systems. Remember to simplify the inequality, isolate the variable, and check the solution. With practice and patience, you can become a master of solving inequalities.