Consider U = {x|x is a real number}. A = {x|x ∈ U and x + 2 > 10}
B = {x|x ∈ U and 2x > 10}

Which pair of statements is true?

5 ∉ A; 5 ∈ B
6 ∈ A; 6 ∉ B
8 ∉ A; 8 ∈ B
9 ∈ A; 9 ∉ B

Explanation:

Let's check choice A

If we plugged x = 5 into the inequality for set A, then,

x+2 > 10

5+2 > 10

7 > 10

which is false. So 5 ∉ A is a true statement. It means "5 is not in set A".

Let's plug x = 5 into the inequality for set B

2x > 10

2*5 > 10

10 > 10

Which is false. So x = 5 is not in set B. The statement 5 ∈ B is false. It should be 5 ∉ B instead.

We can cross choice A off the list.

Now onto choice B

Let's plug x = 6 into the inequality for set A

x+2 > 10

6+2 > 10

8 > 10

This is false, so saying 6 ∈ A is false.

Cross choice B off the list.

Choice C

If we plugged x = 8 into the inequality for set A, then x+2 > 10 would turn into 10 > 10, but that's false. So saying 8 ∉ A is a true statement.

If we plugged x = 8 into the inequality for set B, then we'd go from 2x > 10 to 16 > 10. That being true leads to 8 ∈ B being true.

We conclude that choice C is the final answer since both 8 ∉ A and 8 ∈ B are true statements.

We could stop at choice C, as we already found the answer, but let's check choice D.

Plug x = 9 into the inequality for set A

x+2 > 10

9+2 > 10

11 > 10

So saying 9 ∈ A is true, since x = 9 makes x+2 > 10 true.

Now try x = 9 into set B

2x > 10

2*9 > 10

18 > 10

We see that x = 9 is also in set B. So it should be 9 ∈ B and not 9 ∉ B

In other words, the first part of D is correct, but the second part is not.

We can cross choice D off the list.

question: how many people can she serve?

answer: she can serve 16.66666667 people. so she can only give 16 people a complete serving.

question: does she have enough ice cream for her guests?

answer: no. explanation: she only has enough for 16 people to have a full serving. however, she can give a 17th person a partial serving.