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# What is the product of all positive integers n that satisfy the inequalities 3n + 8 < 30 and 2n - 9 > -4?

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What is the product of all positive integers n that satisfy the inequalities 3n + 8 < 30 and 2n - 9 > -4?

Hey. Pls solve this CPhil or Melody. I need help.

Ok, so I tried to make it into a triple inequality after simplifying like this

2.5 < n < 12.66666667

To get 3,4,5,6,7,8,9,10,11,12.

Mulitplying all those didn't work.

So I tried doing each one individualy and go the same answer.

I dont know where I went wrong.

Aug 16, 2022

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Inequality 1: $$3n + 8< 30$$

Subtract 8 from both sides: $$3n <22$$

Divide by 3: $$n < {22 \over 3}$$

Inequality 2: $$2n - 9 > -4$$

Add 9 to both sides: $$2n > 5$$

Divide both sides by 2: $$n > {5 \over 2}$$

So our range becomes $${5 \over 2} < n < {22 \over 3}$$

The integers that work are 3, 4, 5, 6, and 7.

So the answer is $$3 \times 4 \times 5 \times 6 \times 7 = \color{brown}\boxed{2520}$$

Aug 17, 2022