Question

Use set-builder notation to describe all real numbers satisfying the given conditions.Three times the sum of five and a number is at most 48 The solution in set-builder notation is (xx) (Simplify your answer)

Answer 1

The word problem is translated to set-builder notation by solving the inequality 3(5+x) ≤ 48, which simplifies to x ≤ 11. The set is denoted as x ∈ ℝ , representing all real numbers x that are at most 11.

Step-by-step explanation:

The question involves translating a word problem into set-builder notation. The condition we are given is that 'three times the sum of five and a number is at most 48'. Let's denote the number as 'x'. The sum of five and a number is written as 5+x. The condition 'at most' indicates that the expression should be less than or equal to 48 when we multiply this sum by three, so 3(5+x) ≤ 48.

To find the solution set, we first solve the inequality:

1. 
   
3. Multiply the sum by three: 3(5+x) = 15+3x.
4. 
   
6. Set the inequality: 15+3x ≤ 48.
7. 
   
9. Subtract 15 from both sides: 3x ≤ 48 - 15, which simplifies to 3x ≤ 33.
10. 
    
12. Divide both sides by 3 to isolate x: x ≤ 11.

Now that we have isolated 'x', we can use set-builder notation to define our solution: x ∈ ℝ . This translates to 'the set of all x in the real numbers such that x is less than or equal to 11'.