Solving Inequalities A Step-by-Step Guide

In this comprehensive guide, we will delve into the intricacies of solving inequalities, with a particular focus on the critical steps involved in isolating the variable. Inequalities, unlike equations, express a relationship where one side is not necessarily equal to the other. Instead, they indicate a range of possible values. Understanding inequalities is crucial in various fields, including mathematics, physics, economics, and computer science. This guide aims to provide a clear and concise explanation of the steps involved in solving inequalities, using a specific example to illustrate the process. Let's begin with a fundamental understanding of what inequalities represent and how they differ from equations. Inequalities are mathematical statements that compare two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving an inequality means finding the range of values that satisfy the given relationship. This often involves isolating the variable on one side of the inequality, similar to solving equations, but with some key differences. One of the initial steps in solving inequalities often involves simplifying the expressions on both sides. This can include combining like terms, distributing constants, or, as in our case, subtracting a constant from both sides. The principle behind this step is to maintain the balance of the inequality while moving towards isolating the variable. Subtracting the same value from both sides ensures that the inequality relationship remains valid. For example, if we start with the inequality $5 - 8x < 2x + 3$, subtracting 3 from both sides is a legitimate operation that helps us simplify the inequality and move closer to solving for x. This step is based on the property of inequalities that states: if a < b, then a - c < b - c for any real number c. This property ensures that the direction of the inequality is preserved when subtracting the same value from both sides. Subtracting a constant from both sides of an inequality is a common strategy for simplifying the expression and isolating the variable. This step is often used in conjunction with other algebraic manipulations to solve for the unknown variable. It is crucial to remember that the goal is to isolate the variable while maintaining the integrity of the inequality. Subtracting 3 from both sides of the inequality $5 - 8x < 2x + 3$ leads to a simplified expression that brings us closer to isolating x. By performing this step correctly, we ensure that the solution set for the inequality remains unchanged. This method is widely used in algebra and is an essential skill for solving more complex mathematical problems. The correct application of this step is vital for achieving accurate results in inequality problems. In conclusion, subtracting a constant from both sides of an inequality is a fundamental step in solving for the variable. It is based on the principle of maintaining the balance of the inequality and is a crucial part of the algebraic manipulation process. By understanding and correctly applying this step, we can effectively solve a wide range of inequality problems. In our specific example, this step sets the stage for further simplification and ultimately allows us to determine the solution set for the inequality.

Step 2: Combine like terms and isolate the variable term

After performing the initial step of subtracting 3 from both sides of the inequality, we arrive at a new, simplified form. The next crucial step in solving the inequality is to combine like terms and further isolate the variable term. This step is essential because it consolidates the terms involving the variable, making it easier to isolate the variable in the subsequent steps. Combining like terms is a fundamental algebraic technique that involves adding or subtracting terms with the same variable and exponent. In the context of inequalities, this step simplifies the expression and brings the variable terms closer to one side of the inequality. This process is crucial for isolating the variable and ultimately finding the solution set. The goal of this step is to gather all the terms containing the variable on one side of the inequality and all the constant terms on the other side. This separation is key to isolating the variable and determining the range of values that satisfy the inequality. It involves adding or subtracting terms from both sides of the inequality while maintaining the balance of the relationship. Isolating the variable term often involves using inverse operations to move terms across the inequality sign. For example, if we have an inequality with a variable term added to a constant, we can subtract that constant from both sides to isolate the variable term. Similarly, if a variable term is subtracted from a constant, we can add that constant to both sides. In our example, after subtracting 3 from both sides of $5 - 8x < 2x + 3$, we obtain $2 - 8x < 2x$. The next step is to isolate the x terms. This can be achieved by adding 8x to both sides, resulting in $2 < 10x$. This manipulation effectively moves all the x terms to the right side of the inequality, leaving the constant term on the left. This step is based on the principle that adding or subtracting the same value from both sides of an inequality preserves the relationship. This principle allows us to rearrange terms without changing the solution set of the inequality. By isolating the variable term, we make it easier to determine the range of values that satisfy the inequality. The process of combining like terms and isolating the variable term is a cornerstone of solving inequalities. It involves strategic algebraic manipulations to simplify the expression and bring the variable terms closer to one side of the inequality. This step is often followed by dividing both sides by the coefficient of the variable to fully isolate the variable. The correct execution of this step is vital for arriving at the accurate solution set for the inequality. It demonstrates a solid understanding of algebraic principles and their application in solving mathematical problems. In summary, combining like terms and isolating the variable term is a critical step in solving inequalities. It involves using inverse operations to move terms across the inequality sign while maintaining the balance of the relationship. This step is essential for simplifying the expression and bringing the variable terms closer to one side of the inequality, setting the stage for the final step of dividing by the coefficient of the variable. By mastering this step, we can effectively solve a wide range of inequality problems and gain a deeper understanding of algebraic principles.

Step 3 Divide both sides of the inequality by the coefficient of x

The final and pivotal step in solving an inequality is to divide both sides of the inequality by the coefficient of x. This step isolates the variable, allowing us to determine the solution set of the inequality. However, it is crucial to consider the sign of the coefficient of x, as this will dictate whether the direction of the inequality sign needs to be reversed. Dividing both sides of an inequality by the coefficient of x is the ultimate step in isolating the variable and finding the solution. This process is similar to solving equations, but with a critical distinction: the direction of the inequality sign must be considered when dividing by a negative number. The coefficient of x is the numerical factor that multiplies the variable x. For example, in the term -8x, the coefficient of x is -8. Dividing both sides of the inequality by this coefficient will isolate x on one side, allowing us to determine the range of values that satisfy the inequality. The rule to remember when dividing both sides of an inequality by a negative number is that the direction of the inequality sign must be reversed. This is because multiplying or dividing by a negative number changes the order of the numbers on the number line. For example, if we have -2 < 4, dividing both sides by -2 gives us 1 > -2. The inequality sign has been reversed to maintain the correct relationship. This rule is a fundamental aspect of solving inequalities and must be applied correctly to arrive at the accurate solution set. Failing to reverse the inequality sign when dividing by a negative number will lead to an incorrect solution. Let's consider our example, where we have $2 < 10x$ after step 2. To isolate x, we need to divide both sides by 10, which is the coefficient of x. Since 10 is a positive number, we do not need to reverse the inequality sign. Dividing both sides by 10 gives us $2/10 < x$, which simplifies to $1/5 < x$ or $x > 1/5$. This inequality tells us that x is greater than 1/5, meaning any value greater than 1/5 will satisfy the original inequality. However, if we had an inequality like -10x < 2, dividing both sides by -10 would require us to reverse the inequality sign, resulting in $x > -1/5$. The act of dividing by the coefficient of x is not just a mechanical step; it requires careful consideration of the sign of the coefficient. It is a demonstration of understanding the properties of inequalities and how they differ from equations. The correct application of this step is vital for obtaining accurate solutions in inequality problems. In summary, dividing both sides of the inequality by the coefficient of x is the final step in solving for the variable. It involves isolating x and determining the solution set. However, the crucial point to remember is that the direction of the inequality sign must be reversed when dividing by a negative number. This rule is fundamental to solving inequalities correctly and must be applied with precision. By mastering this step, we can confidently solve a wide range of inequality problems and further develop our algebraic skills.

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What is the missing step in solving the inequality $5-8x < 2x + 3$? Discussion category mathematics

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Solving Inequalities Step-by-Step Guide with Examples