Step by step
The question
Question: Books cost $8 and games cost $20.
Noah must spend at least $100.
Can he buy 5 books and 2 games?
1 Find what we know
What do we know?
Books cost $8 each.
Games cost $20 each.
Noah must spend $100 or more.
What do we want?
To know if 5 books and 2 games fit.
2 Name the unknowns
We don't know how many books.
We call that number x.
We don't know how many games either.
It is a different number, so it gets its own letter: y.
3 Write the inequality
One book costs $8.
So x books cost 8 times x: 8x.
One game costs $20.
So y games cost 20 times y: 20y.
At least $100 means $100 or more.
That is the sign more than or equal to.
In words: the book money plus the game money is at least $100.
So the inequality is 8x + 20y ≥ 100.
Careful!
At least means 100 or more.
So the sign is more than or equal to, not less than.
4 Put the point in
5 books: x = 5.
2 games: y = 2.
As a point, that is (5, 2).
We put them in.
Is it more than or equal to 100?
We don't know yet.
8 times 5 is 40.
20 times 2 is 40.
40 plus 40 is 80.
Is 80 more than or equal to 100?
No.
That is false.
5 Answer the question
No: 5 books and 2 games cost $80.
6 Check
Let's check with the money.
That is $20 short.
So it is not enough.
It works!
Let's look back.
First: find what we know.
Then: name the unknowns.
Then: write the inequality.
Then: put the point in.
Then: answer the question.
Last: check.