Solving Inequalities Finding The Value Of X In 3(x-4) ≥ 5x+2

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In the realm of mathematics, inequalities play a crucial role in defining relationships where quantities are not necessarily equal. Unlike equations that seek to find specific values that make both sides equivalent, inequalities deal with ranges of values that satisfy a given condition. These conditions often involve comparisons like 'greater than' (>), 'less than' (<), 'greater than or equal to' (≥), and 'less than or equal to' (≤). A solution set for an inequality encompasses all the values that, when substituted for the variable, make the inequality statement true. Finding the solution set is a fundamental skill in algebra and is essential for solving real-world problems involving constraints and limitations.

To effectively solve inequalities, a solid understanding of algebraic manipulation is crucial. This includes the ability to distribute, combine like terms, and perform operations on both sides of the inequality while maintaining its balance. However, a critical difference between solving equations and inequalities lies in how we handle multiplication or division by negative numbers. When we multiply or divide both sides of an inequality by a negative number, we must flip the inequality sign to preserve the truth of the statement. For instance, if a > b, then -a < -b. This seemingly simple rule is vital for obtaining correct solutions and avoiding common pitfalls.

Furthermore, the solution set of an inequality can be represented in various ways, including interval notation, set-builder notation, and graphically on a number line. Each representation offers a unique perspective on the range of values that satisfy the inequality. Interval notation uses parentheses and brackets to indicate whether endpoints are included or excluded from the solution set. Set-builder notation provides a concise way to describe the solution set using mathematical symbols and logical conditions. Graphically, the solution set is visualized as a shaded region on the number line, with open circles indicating excluded endpoints and closed circles indicating included endpoints. Mastering these different representations enhances our ability to communicate and interpret solutions effectively.

Our primary goal is to determine which of the provided values for x (-10, -5, 5, or 10) lies within the solution set of the inequality 3(x - 4) ≥ 5x + 2. This requires us to systematically solve the inequality and identify the range of values that satisfy it. The given inequality is a linear inequality, meaning it involves a variable raised to the first power. Solving linear inequalities involves similar steps to solving linear equations, but with the added consideration of the direction of the inequality sign. We'll need to carefully apply algebraic principles to isolate x and determine the solution set.

To begin, we must simplify both sides of the inequality by applying the distributive property and combining like terms. The distributive property allows us to multiply the 3 outside the parentheses by each term inside the parentheses: 3(x - 4) becomes 3x - 12. Our inequality now looks like this: 3x - 12 ≥ 5x + 2. The next step involves rearranging the terms to group the x terms on one side and the constant terms on the other. This is achieved by adding or subtracting terms from both sides of the inequality. Our aim is to isolate x and determine its possible values. By performing these operations carefully, we maintain the balance of the inequality and move closer to the solution.

Once we've isolated x, we'll have a clear understanding of the range of values that satisfy the inequality. The solution set might include all values greater than or equal to a certain number, less than or equal to a certain number, or fall within a specific interval. We can then compare the given options (-10, -5, 5, and 10) to this solution set to determine which one fits. This process of solving the inequality and then checking the provided values is a fundamental approach in algebra. By mastering this technique, we can confidently tackle a wide range of inequality problems.

Now, let's embark on the process of solving the inequality 3(x - 4) ≥ 5x + 2 step by step. As we discussed earlier, the initial step involves applying the distributive property to simplify the left side of the inequality. Multiplying 3 by both x and -4 within the parentheses gives us 3x - 12. This transforms our inequality into 3x - 12 ≥ 5x + 2.

Our next objective is to consolidate the x terms on one side of the inequality and the constant terms on the other. To achieve this, we can subtract 3x from both sides of the inequality. This operation eliminates the x term from the left side and moves it to the right side. Doing so, we get -12 ≥ 2x + 2. It's crucial to perform the same operation on both sides to maintain the balance and validity of the inequality.

Following this, we need to isolate the term containing x. Currently, we have 2x + 2 on the right side. To isolate 2x, we subtract 2 from both sides of the inequality. This gives us -14 ≥ 2x. We are now one step closer to isolating x and determining its possible values.

The final step in isolating x is to divide both sides of the inequality by the coefficient of x, which is 2. Dividing both sides by 2, we obtain -7 ≥ x. This can also be written as x ≤ -7. This is a crucial point: since we divided by a positive number, the direction of the inequality sign remains unchanged. If we had divided by a negative number, we would have needed to flip the sign.

Therefore, the solution set of the inequality 3(x - 4) ≥ 5x + 2 consists of all values of x that are less than or equal to -7. Now we have a clear range of values that satisfy the inequality, and we can proceed to check the given options.

With the solution set of the inequality determined as x ≤ -7, our next task is to examine the provided options (A. -10, B. -5, C. 5, D. 10) and identify which value of x falls within this range. This involves a straightforward comparison of each option with the inequality condition. We are looking for a value that is less than or equal to -7.

Let's consider each option one by one:

  • A. -10: Is -10 less than or equal to -7? Yes, -10 ≤ -7. Therefore, -10 is a potential solution.
  • B. -5: Is -5 less than or equal to -7? No, -5 > -7. So, -5 is not a solution.
  • C. 5: Is 5 less than or equal to -7? No, 5 > -7. Hence, 5 is not a solution.
  • D. 10: Is 10 less than or equal to -7? No, 10 > -7. Thus, 10 is not a solution.

Upon careful examination, we find that only one value, -10, satisfies the inequality x ≤ -7. The other options, -5, 5, and 10, are all greater than -7 and therefore do not belong to the solution set.

In conclusion, after systematically solving the inequality 3(x - 4) ≥ 5x + 2 and comparing the given options with the resulting solution set, we've determined that the correct value of x is A. -10. This value is the only one among the choices that satisfies the inequality x ≤ -7. The other options, -5, 5, and 10, do not fall within the solution set.

This exercise highlights the importance of understanding and applying the principles of solving inequalities. It demonstrates the step-by-step process of simplifying, isolating the variable, and interpreting the solution set. Furthermore, it underscores the need to carefully consider the direction of the inequality sign, especially when multiplying or dividing by negative numbers.

By mastering these skills, we gain the ability to tackle a wide range of algebraic problems involving inequalities and solution sets. This is a fundamental concept in mathematics with applications in various fields, including physics, engineering, and economics. The ability to solve inequalities is essential for modeling real-world scenarios involving constraints and limitations, making it a valuable tool in problem-solving and decision-making.