1. Find the solution set of the linear inequality [tex]x+4 \leq 3x-16x[/tex]. The domain is [tex] \{1,2,3,4,5,7,8,9,10\} [/tex].
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 [tex] \{1,2,3,4,5,6,7,8,9, 10\} [/tex]
Answer (1)
Solve x + 4 ≤ 3 x − 16 x with domain D 1 : Simplify to x ≤ − 7 2 , resulting in an empty set solution ∅ .
Solve 21 − 5 x < 1 for natural numbers: Simplify to 4"> x > 4 , giving the solution set { 5 , 6 , 7 , 8 , 9 , ... } .
For variable k with domain D 3 and no inequality: The solution set is the entire domain { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } .
Explanation
Problem Analysis We are given three problems to solve:
Solve the inequality x + 4 ≤ 3 x − 16 x with the domain D 1 = { 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 } .
Solve the inequality 21 − 5 x < 1 where x is a natural number.
The variable k has a domain D 3 = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } .
Solving Problem 1 Let's solve the first problem. The inequality is x + 4 ≤ 3 x − 16 x . We can simplify this to:
x + 4 ≤ − 13 x
Now, we want to isolate x . Add 13 x to both sides:
14 x + 4 ≤ 0
Subtract 4 from both sides:
14 x ≤ − 4
Divide 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 intersection of the solution set x ≤ − 7 2 and D 1 . Since all the elements in D 1 are positive integers and − 7 2 is a negative number, there are no elements in D 1 that satisfy the inequality. Therefore, the solution set for the first problem is the empty set.
Solving Problem 2 Now, let's solve the second problem. The inequality is 21 − 5 x < 1 . We want to isolate x :
Subtract 21 from both sides:
− 5 x < − 20
Divide by -5 (and remember to flip the inequality sign since we are dividing by a negative number):
4"> x > 4
Since x is a natural number, the solution set is all natural numbers greater than 4. We can write this as x ∈ { 5 , 6 , 7 , 8 , 9 , ... } .
Solving Problem 3 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 to solve for k . Therefore, we cannot determine a specific solution set for k . The solution set is simply the domain itself, i.e., k ∈ { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } .
Final Answer In summary:
The solution set for the inequality x + 4 ≤ 3 x − 16 x with the domain D 1 = { 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 } is the empty set, denoted as ∅ .
The solution set for the inequality 21 − 5 x < 1 where x is a natural number is 4"> x > 4 , which can be written as x ∈ { 5 , 6 , 7 , 8 , 9 , ... } .
For the third problem, since there is no inequality or equation to solve, the solution set for k is the entire domain D 3 = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when planning a budget, you might use inequalities to determine how much you can spend on different items while staying within your income. Similarly, in science, inequalities can help define the range of acceptable values for experimental parameters. In computer science, inequalities are used in algorithms to optimize performance and ensure that certain conditions are met. These examples show how mastering inequalities can provide valuable problem-solving skills across various fields.
