Solve In N The Inequality (x+1)/3 + (x+2)/4 + (x+3)/5 < 5.
In the realm of mathematics, inequalities play a crucial role in defining the boundaries and constraints of variables within a given expression. Solving inequalities involves finding the range of values that satisfy the inequality condition. This article delves into the process of solving a specific inequality within the set of natural numbers (N). The inequality in question is (x+1)/3 + (x+2)/4 + (x+3)/5 < 5. We will explore the step-by-step method to find the solution set for this inequality, ensuring a comprehensive understanding of the underlying principles and techniques.
Before diving into the solution, it's essential to grasp the fundamental concepts of inequalities. An inequality is a mathematical statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving an inequality means determining the values of the variable that make the inequality true. Unlike equations, which have specific solutions, inequalities often have a range of solutions. When dealing with natural numbers (N), we consider only positive integers (1, 2, 3, ...) as potential solutions. This restriction simplifies the solution process, as we can test integer values to see if they satisfy the inequality.
The problem we aim to solve is the inequality (x+1)/3 + (x+2)/4 + (x+3)/5 < 5, where x belongs to the set of natural numbers (N). This means we are looking for positive integers that, when substituted for x, make the left-hand side of the inequality less than 5. The inequality involves fractions and addition, requiring careful algebraic manipulation to isolate x and find the solution set. The solution set will be a range of natural numbers that satisfy the given condition.
Step-by-Step Solution
1. Clearing the Fractions
To simplify the inequality, the first step involves clearing the fractions. This can be achieved by finding the least common multiple (LCM) of the denominators, which are 3, 4, and 5. The LCM of 3, 4, and 5 is 60. Multiplying both sides of the inequality by 60 will eliminate the fractions, making the inequality easier to work with. This is a crucial step in solving inequalities involving fractions, as it transforms the expression into a more manageable form.
Multiplying both sides by 60, we get:
60 * [(x+1)/3 + (x+2)/4 + (x+3)/5] < 60 * 5
This simplifies to:
20(x+1) + 15(x+2) + 12(x+3) < 300
2. Expanding the Terms
Next, we expand the terms to remove the parentheses. This involves distributing the coefficients (20, 15, and 12) across the terms inside the parentheses. Expanding the terms is a fundamental algebraic technique that helps simplify expressions by removing grouping symbols and combining like terms. This step is essential for isolating the variable x and progressing towards the solution.
Expanding the terms, we have:
20x + 20 + 15x + 30 + 12x + 36 < 300
3. Combining Like Terms
Now, we combine the like terms on the left-hand side of the inequality. This involves adding the terms with 'x' and the constant terms separately. Combining like terms simplifies the inequality by reducing the number of terms, making it easier to solve. This step consolidates the expression, bringing us closer to isolating x.
Combining like terms, we get:
(20x + 15x + 12x) + (20 + 30 + 36) < 300
Which simplifies to:
47x + 86 < 300
4. Isolating the Variable
To isolate the variable 'x', we subtract 86 from both sides of the inequality. Isolating the variable is a key step in solving any inequality or equation, as it brings the variable to one side and the constants to the other, allowing us to determine the range of values that satisfy the inequality. This step sets the stage for the final solution.
Subtracting 86 from both sides, we have:
47x < 300 - 86
Which simplifies to:
47x < 214
5. Solving for x
To solve for 'x', we divide both sides of the inequality by 47. Solving for x involves performing the necessary operations to get 'x' by itself on one side of the inequality. This step directly leads to the solution range for x.
Dividing both sides by 47, we get:
x < 214 / 47
Which simplifies to:
x < 4.553 (approximately)
6. Considering Natural Numbers
Since we are looking for solutions in the set of natural numbers (N), we need to consider only positive integers. The inequality x < 4.553 implies that x can be any natural number less than 4.553. Considering natural numbers means restricting the solution set to positive integers only, which is crucial for the specific context of this problem.
Thus, the natural numbers that satisfy the inequality are 1, 2, 3, and 4.
The solution set for the inequality (x+1)/3 + (x+2)/4 + (x+3)/5 < 5 in the set of natural numbers (N) is {1, 2, 3, 4}. This means that when x is 1, 2, 3, or 4, the inequality holds true. The final solution provides the specific values that satisfy the given condition, completing the solution process.
Verification
To ensure the correctness of our solution, we can verify by substituting each value (1, 2, 3, and 4) back into the original inequality:
- For x = 1: (1+1)/3 + (1+2)/4 + (1+3)/5 = 2/3 + 3/4 + 4/5 = 0.667 + 0.75 + 0.8 = 2.217 < 5 (True)
- For x = 2: (2+1)/3 + (2+2)/4 + (2+3)/5 = 3/3 + 4/4 + 5/5 = 1 + 1 + 1 = 3 < 5 (True)
- For x = 3: (3+1)/3 + (3+2)/4 + (3+3)/5 = 4/3 + 5/4 + 6/5 = 1.333 + 1.25 + 1.2 = 3.783 < 5 (True)
- For x = 4: (4+1)/3 + (4+2)/4 + (4+3)/5 = 5/3 + 6/4 + 7/5 = 1.667 + 1.5 + 1.4 = 4.567 < 5 (True)
All the values satisfy the inequality, confirming our solution set.
In this article, we have successfully solved the inequality (x+1)/3 + (x+2)/4 + (x+3)/5 < 5 within the set of natural numbers (N). The solution set {1, 2, 3, 4} represents the values of x that make the inequality true. The process involved clearing fractions, expanding terms, combining like terms, isolating the variable, and considering the natural number constraint. Understanding and applying these steps is crucial for solving various mathematical inequalities. The conclusion of this exercise reinforces the importance of algebraic manipulation and logical reasoning in solving mathematical problems.