The sum of 2 consecutive integers is at most the difference between nine times the smaller and 5 times the large.
x+y <= 9x-5y
6y <= 8x
3y <= 4x
You might also want to consider
x+y <= |9x-5y|
but the integers are consecutive, so
smaller number --- x
next integer ---- x+1
x + x+1 ≤ 9x - 5(x+1)
2x + 1 ≤ 4x - 5
2x ≥ -6
x ≥ -3
Also consider Steve's other possibility .
It usually helps to read the problem carefully.
Nice work, Reiny
To solve this problem, let's first break it down step by step.
Step 1: Understand the problem.
The problem states that the sum of two consecutive integers is at most the difference between nine times the smaller integer and five times the larger integer. We need to find the inequality that represents this relationship.
Step 2: Assign variables.
Let's assign variables to the unknowns in the problem. Let's say the smaller integer is x, and the larger integer is (x + 1) since they are consecutive integers.
Step 3: Translate the problem statement into an inequality.
Given that the sum of the two consecutive integers is less than or equal to the difference between nine times the smaller and five times the larger, we can write the inequality as:
x + (x + 1) ≤ 9x - 5(x + 1)
Step 4: Simplify the inequality.
Let's simplify the inequality by performing the operations on both sides:
2x + 1 ≤ 9x - 5x - 5
Doing the math:
2x + 1 ≤ 4x - 5
Subtracting 4x from both sides:
2x - 4x + 1 ≤ -5
-2x + 1 ≤ -5
Step 5: Isolate the variable.
To isolate the variable, we need to get rid of the constant term on the left side of the inequality. We can do this by subtracting 1 from both sides:
-2x ≤ -5 - 1
-2x ≤ -6
Step 6: Solve for x.
To solve for x, we need to divide both sides of the inequality by -2. When dividing an inequality by a negative number, we need to reverse the sign of the inequality:
x ≥ -6 / -2
x ≥ 3
Step 7: Interpret the solution.
The solution to the inequality is x ≥ 3, which means that the smaller integer must be greater than or equal to 3. Since the problem states that the integers are consecutive, the larger integer must be (x + 1). Therefore, the larger integer must be greater than or equal to 3 + 1, which is 4.
So, the solution to the problem is that the smaller integer must be greater than or equal to 3, and the larger integer must be greater than or equal to 4.