three times an odd integer is eleven less than four times the next greater even integer. What is the odd integer?

3(x+2) = 4x-5

3x+ 6 = 4x -5

Subtract 3x from each side:
3x-3x + 6 = 4x-3x -5
6=x-5

Add +5 to each side:
6+5= x-5+5
11=x
x+2= 13.

let the first odd integer be n

then the next even integer is n+1

3n < 4(n+1) by 11
3n = 4(n+1) - 11
3n = 4n + 4 - 11
7 = n

the first odd integer is 7, then next even one is 8

check:
3 times the first = 21
4 times the second = 32
Is 21 eleven less than 32 ??, YES

Let's solve step-by-step for the odd integer.

Step 1: Let's assume the odd integer is represented by "x".

Step 2: "Three times an odd integer" can be represented as "3x".

Step 3: "Four times the next greater even integer" can be represented as "4(x+2)" since the next greater even integer would be "x+2".

Step 4: "Eleven less than four times the next greater even integer" can be represented as "4(x+2) - 11".

Step 5: Now we have the equation "3x = 4(x+2) - 11".

Step 6: Let's solve for x by simplifying the equation:
3x = 4x + 8 - 11
3x = 4x - 3

Step 7: Moving the variables to one side, we have:
4x - 3x = 3
x = 3

Step 8: Therefore, the odd integer is 3.

So, the odd integer is 3.

To solve this problem, let's break it down into steps:

Step 1: Determine the unknowns
Let's assign variables to the unknowns in the problem:
- Let n be the odd integer (unknown)
- Let n + 2 be the next greater even integer (unknown)

Step 2: Translate the problem into equations
Now let's translate the given information into equations:
- "Three times an odd integer" can be expressed as 3n.
- "Four times the next greater even integer" can be expressed as 4(n + 2).
- "Eleven less than" can be expressed as -11.

So, we can create the equation:
3n = 4(n + 2) - 11

Step 3: Solve the equation
Now, we solve the equation to find the value of n:

3n = 4(n + 2) - 11 (distribute the 4)
3n = 4n + 8 - 11 (simplify inside the parentheses)
3n = 4n - 3 (combine like terms)
-n = -3 (subtract 4n from both sides)
n = 3 (divide both sides by -1)

Step 4: Analyze the solution
The odd integer is n = 3.

Therefore, the odd integer is 3.