Is 4 A Solution To The Inequality (1/2)x - 3 ≤ -3? A Detailed Explanation

In the realm of mathematics, solving inequalities is a fundamental skill. Inequalities, unlike equations, deal with a range of values rather than a single solution. In this article, we will explore the inequality ${\frac{1}{2} x - 3 \leq -3}$ and determine whether the number 4 is indeed a solution. This involves understanding the properties of inequalities and applying basic algebraic principles to arrive at a conclusive answer. We'll delve into the step-by-step process of solving the inequality, verifying the solution, and discussing the broader implications of the result. Understanding how to solve inequalities is crucial not only for academic purposes but also for real-world applications where constraints and ranges of values are essential considerations. This exercise will enhance your problem-solving skills and deepen your understanding of mathematical concepts.

Before we dive into the specifics of the given inequality, let's establish a clear understanding of what inequalities are 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 (≥). Unlike equations, which aim to find specific values that make the expressions equal, inequalities identify a range of values that satisfy the given condition. For instance, an equation like ${x + 2 = 5}$ has a single solution, ${x = 3}$. On the other hand, an inequality like ${x + 2 < 5}$ has a range of solutions, namely all values of ${x}$ that are less than 3. This distinction is crucial because it affects how we approach solving these types of problems. When solving inequalities, we often manipulate the expressions to isolate the variable, just like in equations. However, there's a critical rule to remember: when multiplying or dividing both sides of an inequality by a negative number, 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 ${-x < 2}$, multiplying both sides by -1 gives us ${x > -2}$. Understanding this rule is essential to avoid errors when solving inequalities. Inequalities are used extensively in various fields, including economics, physics, and computer science, to model constraints, optimize processes, and analyze data. They provide a powerful tool for representing real-world situations where quantities are not fixed but fall within a certain range.

To determine whether 4 is a solution to the inequality ${\frac{1}{2} x - 3 \leq -3}$, we first need to solve the inequality itself. Solving an inequality involves isolating the variable on one side of the inequality sign, similar to solving an equation. We begin by adding 3 to both sides of the inequality. This step is crucial because it helps us to simplify the expression and move closer to isolating the variable ${x}$. Adding 3 to both sides of ${\frac{1}{2} x - 3 \leq -3}$ gives us:

$$\frac{1}{2} x - 3 + 3 \leq -3 + 3$$

$$\frac{1}{2} x \leq 0$$

Next, to isolate ${x}$, we need to eliminate the fraction. We can do this by multiplying both sides of the inequality by 2. This step is essential because it removes the coefficient of ${\frac{1}{2}}$ from the variable ${x}$, making it easier to determine the range of values that satisfy the inequality. Multiplying both sides of ${\frac{1}{2} x \leq 0}$ by 2 yields:

$$2 \cdot \frac{1}{2} x \leq 2 \cdot 0$$

$$x \leq 0$$

This result tells us that the solution to the inequality is all values of ${x}$ that are less than or equal to 0. Now that we have solved the inequality, we can proceed to check whether 4 fits within this solution set. This step is crucial to answering the original question of whether 4 is a solution to the given inequality. Understanding the step-by-step process of solving inequalities is fundamental in mathematics and is a skill that is applicable in various contexts.

Now that we have solved the inequality ${\frac{1}{2} x - 3 \leq -3}$ and found that the solution is ${x \leq 0}$, we need to determine whether 4 is a solution. To check if 4 is a solution, we substitute ${x = 4}$ into the original inequality and see if the resulting statement is true. This is a crucial step in verifying the correctness of our solution and ensuring that the value we are testing indeed satisfies the inequality. Substituting ${x = 4}$ into ${\frac{1}{2} x - 3 \leq -3}$ gives us:

$$\frac{1}{2}(4) - 3 \leq -3$$

$$2 - 3 \leq -3$$

$$-1 \leq -3$$

The statement ${-1 \leq -3}$ is false. This is because -1 is greater than -3 on the number line. Therefore, 4 is not a solution to the inequality ${\frac{1}{2} x - 3 \leq -3}$. Our verification process has clearly shown that substituting 4 into the original inequality leads to a false statement, confirming that 4 does not belong to the solution set of the inequality. This step-by-step check is essential in mathematical problem-solving to ensure accuracy and to reinforce the understanding of the concepts involved. Understanding how to verify solutions is as important as knowing how to solve the problem itself. It helps to avoid errors and builds confidence in the correctness of the answer.

In conclusion, after solving the inequality ${\frac{1}{2} x - 3 \leq -3}$, we found that the solution set consists of all values of ${x}$ such that ${x \leq 0}$. By substituting ${x = 4}$ into the original inequality, we determined that the statement ${-1 \leq -3}$ is false. Therefore, the number 4 is not a solution to the given inequality. This exercise highlights the importance of understanding the properties of inequalities, the steps involved in solving them, and the methods for verifying solutions. Solving inequalities is a fundamental skill in mathematics with applications in various fields. The ability to manipulate inequalities, isolate variables, and interpret the results is crucial for problem-solving and decision-making. Furthermore, the process of checking solutions ensures accuracy and reinforces the understanding of the underlying concepts. This article has provided a detailed explanation of how to solve and verify inequalities, using a specific example to illustrate the process. By understanding these concepts and practicing these skills, you can confidently tackle more complex mathematical problems involving inequalities.