1. Find the solution set of the linear inequality [tex]x+4 \leq 3x-16x[/tex]. The domain is {1,2,3,4,5,7,8,9,10}.
2. If [tex]x[/tex] is defined as natural numbers, find the solution set for the inequality [tex]21-5x\ \textless \ 1[/tex].
3. The variable [tex]k[/tex] has a domain {1,2,3,4,5,6,7,8,9,10}.
Answer (2)
Solve the first inequality x + 4 ≤ 3 x − 16 x , which simplifies to x ≤ − 7 2 . The solution set within the domain { 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 } is { } .
Solve the second inequality 21 − 5 x < 1 , which simplifies to 4"> x > 4 . The solution set for natural numbers is { 5 , 6 , 7 , 8 , 9 , ... } .
For the third problem, no inequality is given for k , so the solution set is the domain itself: { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } .
Explanation
Analyzing the Inequalities Let's analyze each inequality separately to find their solution sets within the given domains.
Solving the First Inequality For the first inequality, we have x + 4 ≤ 3 x − 16 x . Simplifying this, we get: x + 4 ≤ − 13 x Adding 13 x to both sides: 14 x + 4 ≤ 0 Subtracting 4 from both sides: 14 x ≤ − 4 Dividing by 14: x ≤ − 14 4 = − 7 2 Since the domain is D 1 = { 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 } , we need to find the elements in D 1 that satisfy x ≤ − 7 2 . Since − 7 2 is approximately -0.2857, and all elements in D 1 are positive integers, there are no elements in D 1 that satisfy the inequality. Therefore, the solution set for the first inequality is the empty set.
Solving the Second Inequality For the second inequality, we have 21 − 5 x < 1 . Solving for x :
21 − 5 x < 1 Subtracting 21 from both sides: − 5 x < − 20 Dividing by -5 (and flipping the inequality sign because we're dividing by a negative number): 4"> x > 4 Since x is defined as natural numbers, the solution set consists of all natural numbers greater than 4. We can represent this as x ∈ { 5 , 6 , 7 , 8 , 9 , ... } .
Analyzing the Third Problem For the third problem, we are given that the variable k has a domain D 3 = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } . However, there is no inequality or equation provided to solve for k . Therefore, we cannot determine a specific solution set for k . The solution set is simply the domain itself, as no further constraints are given.
Final Answer Therefore, the solution set for the first inequality is the empty set, the solution set for the second inequality is all natural numbers greater than 4, and for the third problem, since there is no inequality, the solution set is the domain itself.
Examples
Understanding inequalities is crucial in various real-life situations, such as budgeting, where you need to ensure your expenses are less than your income. For example, if you have a monthly income of $2000 and fixed expenses of $800, you can use the inequality 800"> 2000 − x > 800 to determine how much you can spend on variable expenses ( x ) while staying within your budget. Solving this inequality gives $x < 1200, meaning you can spend less than $1200 on variable expenses. Similarly, in project management, inequalities can help determine the range of time required to complete tasks, ensuring projects are completed within reasonable deadlines.
The solution set for the first inequality is empty, the solution set for the second inequality includes all natural numbers greater than 4, and the solution set for the variable k is its domain itself. Specifically, k can take all values from 1 to 10. Therefore, the summaries are: Solution Set 1: {}, Solution Set 2: {5, 6, 7, 8, 9, 10, ...}, Solution Set 3: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
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