Check the direction of the inequality

Oct 13, 2025 by windailys.com 56 views

Solving Inequalities: A Step-by-Step Guide to Solving the Inequality $\frac{x}{5} - 1 \leq 2 - \frac{x}{10}$

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. In this article, we will focus on solving the inequality $\frac{x}{5} - 1 \leq 2 - \frac{x}{10}$ . Solving inequalities requires a deep understanding of algebraic operations, properties of inequalities, and the ability to manipulate expressions. In this article, we will break down the solution process into manageable steps, making it easier for readers to understand and apply the concepts.

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.

To simplify the inequality, we can start by combining the fractions on the left-hand side and the right-hand side separately.

$\frac{x}{5} - 1 \leq 2 - \frac{x}{10}$

We can rewrite the inequality as:

$\frac{x}{5} - \frac{x}{10} \leq 2 + 1$

Now, we can combine the fractions on the left-hand side and the right-hand side separately.

$\frac{2x}{10} - \frac{x}{10} \leq 3$

Simplifying further, we get:

$\frac{x}{10} \leq 3$

Now that we have simplified the inequality, we can isolate the variable $x$ on one side of the inequality sign. To do this, we can multiply both sides of the inequality by 10.

$\frac{x}{10} \leq 3$

Multiplying both sides by 10, we get:

$x \leq 30$

In this article, we have solved the inequality $\frac{x}{5} - 1 \leq 2 - \frac{x}{10}$ . We started by simplifying the inequality by combining like terms, and then isolated the variable $x$ on one side of the inequality sign. The final solution is $x \leq 30$ . This solution indicates that the value of $x$ must be less than or equal to 30.

When solving inequalities, it's essential to remember the following tips and tricks:

Simplify the inequality: Combine like terms and simplify the inequality to make it easier to solve.
Isolate the variable: Multiply both sides of the inequality by a constant to isolate the variable on one side of the inequality sign.
Check the direction of the inequality: Make sure to check the direction of the inequality sign after multiplying both sides by a constant.

Solving inequalities has numerous real-world applications in various fields, including:

Science: Inequalities are used to model real-world phenomena, such as population growth, chemical reactions, and physical systems.
Engineering: Inequalities are used to design and optimize systems, such as electrical circuits, mechanical systems, and computer networks.
Economics: Inequalities are used to model economic systems, such as supply and demand, inflation, and unemployment.

In conclusion, solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, readers can solve inequalities with confidence. Remember to simplify the inequality, isolate the variable, and check the direction of the inequality sign. With practice and patience, readers can master the art of solving inequalities and apply it to real-world problems.

For readers who want to learn more about solving inequalities, here are some additional resources:

Online tutorials: Websites such as Khan Academy, Mathway, and Wolfram Alpha offer interactive tutorials and exercises on solving inequalities.
Textbooks: Books such as "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by Michael Spivak provide comprehensive coverage of solving inequalities.
Practice problems: Websites such as IXL, Math Open Reference, and Symbolab offer practice problems and exercises on solving inequalities.

In our previous article, we discussed how to solve the inequality $\frac{x}{5} - 1 \leq 2 - \frac{x}{10}$ . Solving inequalities can be a challenging task, but with practice and patience, anyone can master it. 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 the inequality by combining like terms. This will make it easier to isolate the variable on one side of the inequality sign.

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

A: To isolate the variable in an inequality, you can multiply both sides of the inequality by a constant. This will eliminate the fraction and make it easier to solve 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 $ax + b \leq c$ , where $a$ , $b$ , and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

Factor the quadratic expression, if possible.
Set each factor equal to zero and solve for $x$ .
Use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict inequality sign, such as $<, >, \neq$ . A non-strict inequality is an inequality that is written with a non-strict inequality sign, such as $\leq, \geq, =$ . Strict inequalities are used to indicate that the solution is not inclusive, while non-strict inequalities are used to indicate that the solution is inclusive.

Q: How do I determine the direction of the inequality sign?

A: To determine the direction of the inequality sign, you can use the following steps:

Multiply both sides of the inequality by a positive number.
Multiply both sides of the inequality by a negative number.
Check the direction of the inequality sign after multiplying both sides by a negative number.

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: Failing to simplify the inequality can make it difficult to isolate the variable.
Not checking the direction of the inequality sign: Failing to check the direction of the inequality sign can lead to incorrect solutions.
Not considering the domain of the inequality: Failing to consider the domain of the inequality can lead to incorrect solutions.

Solving inequalities can be a challenging task, but with practice and patience, anyone can master it. By following the steps outlined in this article, readers can solve inequalities with confidence. Remember to simplify the inequality, isolate the variable, and check the direction of the inequality sign. With practice and patience, readers can master the art of solving inequalities and apply it to real-world problems.

For readers who want to learn more about solving inequalities, here are some additional resources:

Online tutorials: Websites such as Khan Academy, Mathway, and Wolfram Alpha offer interactive tutorials and exercises on solving inequalities.
Textbooks: Books such as "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by Michael Spivak provide comprehensive coverage of solving inequalities.
Practice problems: Websites such as IXL, Math Open Reference, and Symbolab offer practice problems and exercises on solving inequalities.

Solving inequalities is a fundamental concept in mathematics that has numerous real-world applications. By following the steps outlined in this article, readers can solve inequalities with confidence. Remember to simplify the inequality, isolate the variable, and check the direction of the inequality sign. With practice and patience, readers can master the art of solving inequalities and apply it to real-world problems.