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

In this mathematical exploration, we delve into the realm of inequalities, specifically focusing on determining the value of x that satisfies the inequality 3(x - 4) ≥ 5x + 2. Inequalities, unlike equations, represent a range of possible solutions rather than a single value. To decipher this inequality, we embark on a step-by-step journey, employing algebraic techniques to isolate x and pinpoint the solution set. This problem not only tests our algebraic manipulation skills but also our understanding of how inequalities behave under various operations. The solution will reveal which of the given options – A. -10, B. -5, C. 5, or D. 10 – falls within the solution set of the inequality. Our primary goal is to methodically solve the inequality, ensuring we maintain the integrity of the inequality throughout the process. This involves distributing, combining like terms, and carefully handling the inequality sign when multiplying or dividing by negative numbers. By mastering these techniques, we can confidently navigate a wide range of inequality problems, a crucial skill in various mathematical and real-world applications. The ability to solve inequalities is not just about finding an answer; it's about understanding the range of values that satisfy a given condition, a concept that resonates in fields like optimization, economics, and physics. So, let's embark on this mathematical journey, unraveling the intricacies of the inequality and discovering the value of x that lies within its solution set. This exploration will not only enhance our problem-solving skills but also deepen our appreciation for the power and elegance of algebraic manipulation.

To solve the inequality 3(x - 4) ≥ 5x + 2, we will proceed systematically, applying algebraic principles to isolate x. The first step in this process involves distributing the 3 on the left side of the inequality. This means multiplying 3 by both x and -4 within the parentheses. This distribution is crucial as it allows us to remove the parentheses and simplify the expression, paving the way for further manipulation. Once we've distributed, the inequality takes a new form, revealing the terms that need to be combined and rearranged. This step is not just about arithmetic; it's about transforming the inequality into a more manageable state, one where we can clearly see the path to isolating x. The careful execution of this distributive property is paramount to the accuracy of the final solution. A mistake in this initial step can cascade through the rest of the process, leading to an incorrect answer. Therefore, we must approach this distribution with precision and attention to detail. After distributing, we'll have a clearer picture of the relationship between the terms involving x and the constant terms. This clarity is essential for the subsequent steps, where we will strategically combine like terms and move them to opposite sides of the inequality. The distributive property, a fundamental concept in algebra, is the key that unlocks the initial complexity of this inequality, allowing us to proceed towards a solution with confidence and understanding. By mastering this step, we lay a solid foundation for tackling more complex inequalities in the future.

1. Distribute: Begin by distributing the 3 on the left side of the inequality: 3x - 12 ≥ 5x + 2.

   This distribution is a fundamental step in simplifying the inequality. By multiplying 3 by both x and -4, we eliminate the parentheses, making the expression easier to manipulate. The result, 3x - 12 ≥ 5x + 2, is a crucial stepping stone towards isolating x. This step exemplifies the importance of the distributive property in algebra, a property that allows us to break down complex expressions into simpler components. The accuracy of this distribution is paramount; any error here will propagate through the rest of the solution, leading to an incorrect answer. Therefore, meticulous attention to detail is essential when applying the distributive property. Once the distribution is complete, we have a clearer view of the terms involving x and the constant terms, setting the stage for the next phase of the solution. This step is not merely about arithmetic; it's about strategically transforming the inequality into a form that is more amenable to algebraic manipulation. By mastering the distributive property, we empower ourselves to tackle a wider range of algebraic problems with confidence and precision. The careful execution of this step lays the foundation for a successful solution, demonstrating the power of fundamental algebraic principles in problem-solving.

2. Combine like terms: Next, we want to gather the x terms on one side of the inequality and the constants on the other. Subtract 3x from both sides: -12 ≥ 2x + 2

   Subtracting 3x from both sides is a strategic move designed to consolidate the x terms on one side of the inequality. This step is crucial for isolating x and ultimately determining the solution set. The principle behind this operation is that subtracting the same quantity from both sides of an inequality maintains the inequality's validity. This ensures that we are transforming the inequality without altering its fundamental relationship. By performing this subtraction, we effectively shift the x terms to the right side of the inequality, paving the way for further simplification. The result, -12 ≥ 2x + 2, represents a significant step forward in our quest to solve for x. This step demonstrates the power of algebraic manipulation in simplifying complex expressions and revealing the underlying relationships between variables. The careful execution of this subtraction is paramount to the accuracy of the solution. A mistake in this step can lead to an incorrect isolation of x and a flawed solution set. Therefore, we must approach this operation with precision and attention to detail. By mastering this technique of combining like terms, we enhance our ability to solve a wide range of inequalities and algebraic equations. This skill is not just about finding an answer; it's about understanding the principles of algebraic manipulation and applying them strategically to solve problems.

3. Isolate the variable: Subtract 2 from both sides: -14 ≥ 2x. Now, divide both sides by 2: -7 ≥ x.

   Subtracting 2 from both sides is another crucial step in isolating x. This operation aims to consolidate the constant terms on one side of the inequality, further simplifying the expression. Similar to the previous step, this subtraction is based on the principle that performing the same operation on both sides of an inequality preserves its validity. This ensures that we are transforming the inequality without altering its fundamental relationship. By subtracting 2, we move closer to isolating the term containing x, which is our primary goal. The result, -14 ≥ 2x, represents a significant advancement in our quest to solve for x. This step highlights the importance of strategic algebraic manipulation in simplifying complex expressions and revealing the underlying relationships between variables. The careful execution of this subtraction is paramount to the accuracy of the solution. A mistake in this step can lead to an incorrect isolation of x and a flawed solution set. Therefore, we must approach this operation with precision and attention to detail. Next, dividing both sides by 2 is the final step in isolating x. This operation aims to completely separate x from any coefficients or constants, revealing its value or range of values. Dividing by a positive number, in this case 2, preserves the direction of the inequality. This is a crucial point, as dividing by a negative number would require us to flip the inequality sign. The result, -7 ≥ x, represents the solution to the inequality. This solution indicates that x is less than or equal to -7. This step demonstrates the power of algebraic manipulation in completely isolating a variable and determining its value or range. The careful execution of this division is paramount to the accuracy of the solution. A mistake in this step can lead to an incorrect determination of the solution set. Therefore, we must approach this operation with precision and attention to detail. By mastering this technique of isolating variables, we enhance our ability to solve a wide range of inequalities and algebraic equations. This skill is not just about finding an answer; it's about understanding the principles of algebraic manipulation and applying them strategically to solve problems.

The solution to the inequality is x ≤ -7. Now, we need to determine which of the given options satisfies this condition. Option A is -10. Since -10 is less than -7, it is in the solution set.

The solution set x ≤ -7 represents a range of values that satisfy the original inequality. This range includes all numbers less than or equal to -7. To determine which of the given options falls within this solution set, we must compare each option to -7. This comparison is a crucial step in verifying our solution and ensuring that we select the correct answer. Each option represents a potential value of x, and we need to ascertain whether that value satisfies the condition x ≤ -7. This involves understanding the number line and the relative positions of numbers. Numbers to the left on the number line are smaller, and numbers to the right are larger. Therefore, any number to the left of -7 on the number line will be less than -7 and thus belong to the solution set. This comparison process is not merely about checking a box; it's about understanding the meaning of the solution set and how it relates to the possible values of x. By carefully evaluating each option, we reinforce our understanding of inequalities and solution sets. This skill is essential for solving a wide range of mathematical problems, including those involving optimization, constraints, and real-world scenarios. The ability to interpret a solution set and identify values that belong to it is a fundamental aspect of mathematical literacy. So, let's meticulously compare each option to -7, ensuring that we select the value that truly satisfies the inequality x ≤ -7.

Therefore, the correct answer is A. -10, as it is the only option that satisfies the inequality x ≤ -7.

The final answer, A. -10, represents the culmination of our step-by-step solution process. This answer is not just a number; it's the value of x that satisfies the given inequality, 3(x - 4) ≥ 5x + 2. This conclusion is reached through a series of logical and algebraic steps, each building upon the previous one. We began by distributing, then combined like terms, and finally isolated x to arrive at the solution set x ≤ -7. This solution set defines a range of values, and -10 is the only option provided that falls within this range. The selection of A. -10 is not arbitrary; it's a direct consequence of the mathematical operations we performed and the logical deductions we made. This answer underscores the importance of precision and accuracy in algebraic problem-solving. Each step, from the initial distribution to the final comparison, must be executed with care to ensure the validity of the solution. The final answer is not just an end point; it's a testament to the power of algebraic manipulation and the ability to solve complex problems through systematic reasoning. By arriving at this answer, we not only solve the specific problem at hand but also reinforce our understanding of inequalities and solution sets. This understanding is a valuable asset in tackling a wide range of mathematical challenges, both in academic settings and in real-world applications. So, let's embrace the final answer, A. -10, as a symbol of our problem-solving prowess and our ability to navigate the intricacies of algebraic inequalities.