Solving and Understanding the Inequality 3x - 5/6 ≥ -2x/3

In the world of algebra, inequalities are just as crucial as equations. An inequality like 3x - 5/6 ≥ -2x/3 challenges us to find the values of x that satisfy this condition. Understanding and solving such inequalities is fundamental to mastering algebraic concepts.

Understanding the Inequality

The inequality we are dealing with is: 3x - 5/6 ≥ -2x/3. This is a linear inequality, and solving it involves algebraic manipulation to isolate the variable x on one side of the inequality.

Solving the Inequality Step-by-Step

Start with the given inequality:

3x - 5/6 ≥ -2x/3

Multiplying both sides by 6 to eliminate the fraction 5/6:

6(3x - 5/6) ≥ 6(-2x/3)

22x - 5 ≥ -4x

Add 4x to both sides to combine the x terms:

26x - 5 ≥ 0

26x ≥ 5

Divide both sides by 26 to isolate x:

x ≥ 5/26

Now that we have the solution, we can interpret it in different contexts. Let's break it down further:

Positive Integers

For positive integers, the inequality has a specific range of solutions. The smallest positive integer that satisfies x ≥ 5/26 is 1 because 1 is the smallest integer greater than 5/26. Therefore, in the context of positive integers, the answer is yes, as there are integers that satisfy this inequality.

General Real Numbers

When considering all real numbers, the solution set is more general. The solution set is x ≥ 5/26. This means any real number x that is greater than or equal to 5/26 satisfies the inequality. Thus, for any real number x in the range [5/26, ∞), the inequality holds true.

Numbers Less Than 5/26

For real numbers less than 5/26, the inequality does not hold. The solution set for these numbers is x

Conclusion

The inequality 3x - 5/6 ≥ -2x/3 has a solution set x ≥ 5/26. For positive integers, the answer is yes because there are integers in this range. For any real number in the range [5/26, ∞), the inequality is satisfied. However, for all real numbers in the range (-∞, 5/26), the inequality does not hold true.

Understanding and solving such inequalities is a fundamental skill in algebra, providing a solid foundation for more complex mathematical concepts.

Related Keywords

inequality, algebraic manipulation, real numbers, solution set