Points to Remember:
- Solving linear inequalities.
- Understanding natural numbers.
- Expressing the solution set.
Introduction:
This question requires solving a linear inequality to find the solution set A, where x represents natural numbers (positive integers). A linear inequality is an inequality that involves a linear expression (a variable raised to the power of 1). Solving it involves manipulating the inequality to isolate the variable, similar to solving linear equations, but with careful consideration of the inequality sign. Natural numbers are the set of positive integers: {1, 2, 3, 4, …}.
Body:
Solving the Inequality:
The given inequality is 9x + 2 > 7x â 2. To solve this, we follow these steps:
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Subtract 7x from both sides: 9x – 7x + 2 > 7x – 7x â 2 which simplifies to 2x + 2 > -2
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Subtract 2 from both sides: 2x + 2 – 2 > -2 – 2 which simplifies to 2x > -4
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Divide both sides by 2: 2x/2 > -4/2 which simplifies to x > -2
Determining the Solution Set A:
Since x represents natural numbers, we are only interested in positive integers. The inequality x > -2 means that x can be any number greater than -2. Considering that x must be a natural number, the solution set A includes all natural numbers greater than -2. This means A = {1, 2, 3, 4, …}.
Conclusion:
The solution to the inequality 9x + 2 > 7x â 2, where x is a natural number, is x > -2. Since x must be a natural number, the solution set A is the set of all positive integers. Therefore, A = {1, 2, 3, 4, …}. This demonstrates a straightforward application of solving linear inequalities within the context of a specific number set. The solution highlights the importance of carefully considering the domain of the variable when solving mathematical problems. This approach ensures accuracy and a complete understanding of the problem’s solution.