Solving The Inequality 0>-3x-2x A Step-by-Step Guide

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In mathematics, inequalities play a crucial role in defining ranges and conditions for variables. Solving inequalities involves finding the set of values that satisfy a given inequality. This article delves into the process of solving the inequality $0>-3x-2x$, providing a detailed, step-by-step solution and explaining the underlying concepts. Understanding how to manipulate and simplify inequalities is fundamental not only in algebra but also in various real-world applications, such as optimization problems, economics, and engineering.

Understanding Inequalities

Before diving into the specific solution, it's important to understand what inequalities represent. Unlike equations, which state that two expressions are equal, inequalities express a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The basic inequality symbols include:

  • >{>}: Greater than
  • <{<}: Less than
  • ≥{\geq}: Greater than or equal to
  • ≤{\leq}: Less than or equal to

When solving inequalities, the goal is to isolate the variable on one side of the inequality symbol, similar to solving equations. However, there is one critical difference: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol. This rule is essential for accurately solving inequalities.

Basic Principles for Solving Inequalities

To effectively solve inequalities, certain principles must be followed. These principles ensure that the solutions obtained are accurate and that the inequality remains balanced throughout the solving process. Understanding these principles is key to mastering the art of solving mathematical problems that involve inequalities.

  • Addition and Subtraction: You can add or subtract the same number from both sides of an inequality without changing its direction. This is because adding or subtracting a constant shifts the entire inequality along the number line, maintaining the relative order of the expressions. For instance, if you have the inequality x+3>5{x + 3 > 5}, you can subtract 3 from both sides to isolate x{x}, resulting in x>2{x > 2}. This principle allows for the simplification of complex inequalities by isolating the variable term.

  • Multiplication and Division by a Positive Number: Multiplying or dividing both sides of an inequality by a positive number does not change the direction of the inequality. This is because positive numbers maintain the order of the expressions. For example, if you have the inequality 2x<6{2x < 6}, you can divide both sides by 2 to solve for x{x}, yielding x<3{x < 3}. The direction of the inequality remains unchanged because the operation involves a positive number.

  • Multiplication and Division by a Negative Number: This is a critical rule in solving inequalities: when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality. This reversal is necessary because multiplying or dividing by a negative number reflects the numbers across the number line, which changes their order. For instance, if you have the inequality −3x>9{-3x > 9}, dividing both sides by -3 requires you to flip the inequality sign, resulting in x<−3{x < -3}. Ignoring this rule can lead to incorrect solutions.

  • Simplifying Expressions: Before applying any operations to isolate the variable, it is often necessary to simplify the expressions on both sides of the inequality. This can involve combining like terms, distributing multiplication over parentheses, or applying other algebraic simplifications. Simplifying the inequality first makes it easier to see the next steps and reduces the chances of making errors. For example, if you have the inequality 2(x+1)−3x<5{2(x + 1) - 3x < 5}, you should first distribute the 2, combine like terms, and then proceed with isolating x{x}.

These principles are the foundation for solving inequalities accurately. By understanding and applying them correctly, you can navigate through various types of inequalities and find the solutions that satisfy the given conditions. Mastering these concepts is essential for anyone studying algebra and beyond, as inequalities are used extensively in higher mathematics and real-world applications.

Step-by-Step Solution to $0>-3x-2x$

Let's solve the inequality $0>-3x-2x$ step by step:

  1. Combine Like Terms: First, combine the like terms on the right side of the inequality. 0>−3x−2x{ 0 > -3x - 2x } 0>−5x{ 0 > -5x }

  2. Isolate the Variable: To isolate x{x}, divide both sides of the inequality by -5. Remember to reverse the direction of the inequality symbol because we are dividing by a negative number. 0−5<−5x−5{ \frac{0}{-5} < \frac{-5x}{-5} } 0<x{ 0 < x }

  3. Rewrite the Inequality: We can rewrite the inequality for clarity. x>0{ x > 0 }

Therefore, the solution to the inequality $0>-3x-2x$ is x>0{x > 0}.

Detailed Steps and Explanation

To fully grasp the solution, let's break down each step in detail. This section aims to provide a comprehensive understanding of the process of solving inequalities, ensuring that readers can apply these methods to similar problems with confidence. Each step is explained with clarity, making the solution accessible to learners of all levels.

Combining Like Terms

The first step in solving inequalities, as in solving equations, is often to simplify both sides of the inequality. This usually involves combining like terms, which are terms that contain the same variable raised to the same power. In the given inequality, $0 > -3x - 2x$, the terms extit{-3x} and extit{-2x} are like terms because they both contain the variable x{x} raised to the power of 1. Combining these terms involves adding their coefficients.

  • Identifying Like Terms: Look for terms that have the same variable part. In this case, both terms have x{x}.
  • Adding Coefficients: Add the coefficients of the like terms. Here, the coefficients are -3 and -2. So, −3+(−2)=−5{-3 + (-2) = -5}.
  • Simplifying the Inequality: Rewrite the inequality by combining the like terms. The inequality $0 > -3x - 2x$ simplifies to $0 > -5x$. This simplification makes the inequality easier to work with in subsequent steps.

Isolating the Variable

After simplifying the inequality by combining like terms, the next step is to isolate the variable. Isolating the variable means getting the variable alone on one side of the inequality. This is achieved by performing operations on both sides of the inequality that move all other terms away from the variable. In this case, we need to isolate x{x} in the inequality $0 > -5x$.

  • Identifying the Operation Needed: Determine what operation is needed to isolate the variable. Here, x{x} is being multiplied by -5, so we need to divide both sides by -5.
  • Applying the Operation: Divide both sides of the inequality by -5. This step is crucial, and it's important to remember the rule about dividing by a negative number. 0−5>−5x−5{ \frac{0}{-5} > \frac{-5x}{-5} }
  • Reversing the Inequality Sign: Because we are dividing by a negative number, we must reverse the direction of the inequality sign. The >{>} sign becomes a <{<} sign. This gives us: 0<x{ 0 < x }
  • Simplifying: Simplify the resulting expressions. 0−5{\frac{0}{-5}} simplifies to 0, and −5x−5{\frac{-5x}{-5}} simplifies to x{x}. The inequality is now 0<x{0 < x}.

Rewriting the Inequality

While 0<x{0 < x} is a correct solution, it is often clearer and more conventional to write the inequality with the variable on the left side. This step involves simply rewriting the inequality so that x{x} comes first. The inequality 0<x{0 < x} states that 0 is less than x{x}, which is the same as saying x{x} is greater than 0.

  • Understanding the Meaning: Recognize that 0<x{0 < x} and x>0{x > 0} convey the same information. Both inequalities mean that x{x} can be any number greater than 0.
  • Rewriting for Clarity: Rewrite the inequality with the variable on the left side. This gives us: x>0{ x > 0 }

This final form, x>0{x > 0}, is the most common way to express the solution to the inequality. It clearly states that the solution set includes all numbers greater than 0. This step ensures that the solution is presented in a way that is easily understood and widely accepted in mathematical notation.

By following these detailed steps, anyone can solve inequalities similar to $0 > -3x - 2x$. The key is to understand the principles behind each step, especially the rule about reversing the inequality sign when multiplying or dividing by a negative number. With practice, these techniques become second nature, allowing for the efficient and accurate solution of a wide range of inequality problems.

Conclusion

In conclusion, solving the inequality $0>-3x-2x$ involves combining like terms, isolating the variable, and remembering to reverse the inequality sign when dividing by a negative number. The solution to this inequality is x>0{x > 0}, which means any value of x{x} greater than 0 will satisfy the original inequality. This step-by-step approach provides a clear methodology for solving similar inequalities, reinforcing the importance of understanding the fundamental principles of algebra. Mastering these techniques is crucial for success in mathematics and its applications in various fields.

Final Answer

The final answer is x>0\boxed{x>0}