WHICH IS BIGGER, n+10 or 2n+3
Let's first say that n+10 is greater than 2n+3. Then the equation goes:
n+10 > 2n+3
then, subtract n from each side:
10 > n+3
subtract 3 from each side:
7 > n
This shows that if n is smaller than 7, n+10 is greater than 2n+3. This fits the example of when n = 4, n+10 is greater than 2n+3, since 4 is smaller than 7.
Now, let's look at is 2n+3 is greater than n+10. Then the equation will go:
2n+3 > n+10
like the previous equation, subtract n from each side:
n+3 >10
and subtract 3:
n > 7
So, this shows that if n is greater than 7, 2n+3 is greater than n+10. This also fits the example of when n = 10, since 10 is greater than 7.
Graphical Solution:
As you see, the red line is the linear equation shows n+10, and the black line shows 2n+3.
The two lines meet at 7, where 2n+3 starts to get larger than n+10. So if n exceeds 7, 2n+3 will be larger than n+10.
OTHER EQUATIONS
Simplifying the problem like this makes the problem easier to solve.
Now let's try some other equations:
2n+7 and 4n+11
Let’s say 2n+7 is greater than 4n+11
2n+7 > 4n+11
Subtract 2n:
7 > 2n+11
Subtract 11:
-4 > 2n
Divide by 2:
-2 > n
So, n has to be less than -2 in order for 2n+7 to be greater than 4n+11, and when 4n+11 is greater than 2n+7:
4n+11 > 2n+7
Subtract 2n:
2n+11 > 7
Subtract 11:
2n+0 > -4
Divide by 2:
n > -2
Now we can see if n is greater than -2, 4n+11 is greater than 2n+7.
Graphical Solution:
The red line represents 4n+11, and the black line represents 2n+7. You can see as n reaches -2, 4n+11 gets greater than 2n+7.
2. 2(3n+4) and 3(2n+4)
Let's first simplify 2(3n+4) and 3(2n+4) to 6n+8 and 6n+12
If 6n+8 is greater than 6n+12:
6n+8 > 6n+12
Divide by 2:
3n+4 > 3n+6
Subtract 3n:
4 > 6
If 6n+8 is greater than 6n+12, we get an equation of 4 is greater than 6, which is totally wrong. So, this proves that 6n+8 cannot be greater than 6n+12, and 6n+12 is always greater than 6n+8, so matter what the value of n is.
Graphical Solution:
In the graph, the black line is 3(2n+4), and the red line is 2(3n+4). We can see two parallel lines, where the value of the black line is always greater than the red line.
3. 2(3n+3) and 3(2n+2)
Let’s simplify this to 6n+6 and 6n+6.
We can see that these two are equal in value, so no matter what n will be, these two equations have the same output.
Graphical Solution:
In the graph, the two lines( red and black) have overlapped. This shows that they are equal in value.