Solving Inequalities The Value Of X In 2(3x - 1) ≥ 4x - 6

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This article delves into the process of solving the inequality 2(3x - 1) ≥ 4x - 6. We will break down the steps involved in isolating the variable x and determining the solution set. Understanding how to solve inequalities is a fundamental skill in algebra, with applications in various fields of mathematics and beyond. This exploration will provide a clear and concise method for tackling similar problems, enhancing your problem-solving capabilities.

Breaking Down the Inequality

To solve the inequality 2(3x - 1) ≥ 4x - 6, the initial step involves distributing the 2 on the left side of the inequality. This process eliminates the parentheses and simplifies the expression, allowing us to further isolate the variable x. The distributive property states that a(b + c) = ab + ac. Applying this property to our inequality, we multiply 2 by both terms inside the parentheses: 3x and -1. This transforms the inequality into a more manageable form, setting the stage for subsequent steps in solving for x. After distributing, the inequality becomes 6x - 2 ≥ 4x - 6. This transformation is crucial because it allows us to combine like terms and ultimately isolate x on one side of the inequality.

Step-by-Step Distribution

  1. Identify the distributive property: Recognize that 2(3x - 1) means 2 multiplied by the entire expression (3x - 1).
  2. Multiply 2 by 3x: 2 * 3x equals 6x.
  3. Multiply 2 by -1: 2 * -1 equals -2.
  4. Rewrite the inequality: Replace 2(3x - 1) with 6x - 2, resulting in the inequality 6x - 2 ≥ 4x - 6.

This distribution step is a foundational algebraic manipulation. It is important to perform it accurately, as any error here will propagate through the rest of the solution. By carefully applying the distributive property, we ensure that we are working with an equivalent inequality that is easier to solve.

Isolating the Variable x

After distributing, we have the inequality 6x - 2 ≥ 4x - 6. The next crucial step is to isolate the variable x. This involves rearranging the terms in the inequality so that all terms containing x are on one side and all constant terms are on the other side. To achieve this, we will perform a series of algebraic manipulations, ensuring that each operation maintains the balance of the inequality. This process is analogous to solving an equation, with the key difference being that we must pay attention to how certain operations affect the inequality sign.

Combining Like Terms

  1. Subtract 4x from both sides: To move the x term from the right side to the left side, we subtract 4x from both sides of the inequality. This gives us 6x - 4x - 2 ≥ 4x - 4x - 6, which simplifies to 2x - 2 ≥ -6. Subtracting the same value from both sides maintains the inequality.
  2. Add 2 to both sides: To move the constant term from the left side to the right side, we add 2 to both sides of the inequality. This gives us 2x - 2 + 2 ≥ -6 + 2, which simplifies to 2x ≥ -4. Adding the same value to both sides also maintains the inequality.

By performing these two operations, we have successfully isolated the x term on the left side of the inequality. We are now one step closer to solving for x. The resulting inequality, 2x ≥ -4, is much simpler to work with and allows us to easily determine the solution set.

Solving for x

Having isolated the x term, we now have the inequality 2x ≥ -4. The final step in solving for x is to divide both sides of the inequality by the coefficient of x, which in this case is 2. This operation will give us the value or range of values for x that satisfy the original inequality. It is crucial to remember that when dividing (or multiplying) both sides of an inequality by a negative number, we must reverse the direction of the inequality sign. However, since we are dividing by a positive number (2), we do not need to reverse the inequality sign in this instance.

Dividing to Find the Solution

  1. Divide both sides by 2: To isolate x, we divide both sides of the inequality 2x ≥ -4 by 2. This gives us (2x)/2 ≥ (-4)/2.
  2. Simplify: Performing the division, we get x ≥ -2. This is the solution to the inequality.

The solution x ≥ -2 indicates that any value of x that is greater than or equal to -2 will satisfy the original inequality 2(3x - 1) ≥ 4x - 6. This solution set includes -2, as well as all numbers to the right of -2 on the number line. Understanding this solution is crucial for answering the question and identifying the correct answer choice.

Identifying the Correct Answer

We have determined that the solution to the inequality 2(3x - 1) ≥ 4x - 6 is x ≥ -2. Now, we need to examine the given answer choices and identify which value of x falls within this solution set. The answer choices are:

  • A. -10
  • B. -5
  • C. -3
  • D. -1

Evaluating the Answer Choices

  1. Check -10: Is -10 ≥ -2? No. -10 is less than -2, so -10 is not in the solution set.
  2. Check -5: Is -5 ≥ -2? No. -5 is less than -2, so -5 is not in the solution set.
  3. Check -3: Is -3 ≥ -2? No. -3 is less than -2, so -3 is not in the solution set.
  4. Check -1: Is -1 ≥ -2? Yes. -1 is greater than -2, so -1 is in the solution set.

Therefore, the correct answer is D. -1, as it is the only value among the choices that satisfies the inequality x ≥ -2.

Conclusion

In summary, solving the inequality 2(3x - 1) ≥ 4x - 6 involves several key steps. First, we distribute the 2 on the left side of the inequality. Second, we isolate the variable x by combining like terms and moving them to the appropriate sides of the inequality. Third, we solve for x by dividing both sides of the inequality by the coefficient of x. Finally, we identify the correct answer by determining which of the given values satisfies the solution set.

By following these steps, we have successfully determined that the value of x in the solution set is -1 (Answer D). This process highlights the importance of understanding and applying algebraic principles to solve inequalities effectively. Mastering these skills is crucial for success in mathematics and related fields.

This comprehensive approach to solving inequalities provides a solid foundation for tackling more complex problems. The ability to manipulate algebraic expressions and understand the properties of inequalities is a valuable asset in problem-solving and critical thinking. By practicing and applying these techniques, you can enhance your mathematical proficiency and build confidence in your ability to solve a wide range of problems.