Step by step
The question
Question: Lily has $35.
She saves $10 each month.
She wants at least $120.
How many months must she save?
1 Find what we know
What do we know?
At least $120.
$10 for each month.
At the start: $35.
What do we want?
How many months.
2 Name the unknown
We don't know how many months.
So we call it x.
At least $120 means $120 or more.
We write ≥ $120.
3 Write the inequality
In words: what the months add, plus the start, is at least $120.
So the inequality is 10x + 35 ≥ 120.
Now we solve it.
We want x by itself.
Two things are stuck to x: times 10 and + 35.
The + 35 is on the outside, so it comes off first.
Then the times 10.
4 Remove the + 35
To remove + 35, we take away 35.
We do it on both sides, so the sign stays true.
+ 35 and − 35 make zero.
Cross them out.
120 − 35 = 85.
Now we have 10x ≥ 85.
5 Remove the 10
10x means 10 times x.
We want just x.
To remove times 10, we divide by 10.
We do it on both sides, so the sign stays true.
10 divided by 10 is 1.
So just x is left.
We can make 85/10 simpler.
It is the same as 17/2.
So x is greater than or equal to 17/2.
6 Show it on a number line
7 Answer the question
17/2 is 8.5.
Months come in whole numbers.
The smallest whole number from 8.5 up is 9.
So we round up.
Careful!
At least means we may not stop short.
So we always round up.
So at least 9 months.
8 Check
Let's check in the story.
9 months: $125.
That is at least $120.
But 8 months would be $115: not enough.
So 9 is the fewest.
So the answer is right!
Let's look back.
First: find what we know.
Then: name the unknown.
Then: write the inequality.
Then: solve it.
Then: answer the question.
Last: check.