THEOREM: n+10 is larger than 2n+3 for values of n from 0 to 6. - They are equal at n=7. 2n+3 becomes larger than n+10 when n is 8 or greater.
Below there is evidence to prove this theorem.
They have got different answers to which equation has the largest total due to the number they used.
n+10=2n+3
n=1
1+10=11
2 times by 1=2+3=5
So n+10 is larger than 2n+3
n+10=2n+3
n=2
2+10=12
2 times by 2=4+3=7
So n+10 is larger than 2n+3
n+10=2n+3
n=3
3+10=13
2 times by 3=6+3=9
So n+10 is larger than 2n+3
n+10=2n+3
n=4
4+10=14
2 times by 4=8+3=11
So n+10 is larger than 2n+3
n+10=2n+3
n=5
5+10=15
2 times by 5=10+3=13
So n+10 is larger than 2n+3
n+10=2n+3
n=6
6+10=16
2 times by 6=12+3=15
So n+10 is larger than 2n+3
n+10=2n+3
n=7
7+10=17
2 times by 7=14+3=17
So both the equations are the same
n+10=2n+3
n=8
8+10=18
2 times by 8=16+3=19
So 2n+3 is larger than n+10
n+10=2n+3
n=9
9+10=19
2 times by 9=18+3=21
So 2n+3 is larger than n+10
n+10=2n+3
n=10
10+10=20
2 times by 10=20+3=23
So 2n+3 is larger than n+10
For the following pairs of expressions, can you work out when each expression is bigger?
2n+7=4n+11
n=1
2 times by 1=2+7=9
4 times by 1=4+11=15
So 4n+11 is larger than 2n+7
2n+7=4n+11
n=2
2 times by 2=4+7=11
4 times by 2=8+11=19
So 4n+11 is larger than 2n+7
2n+7=4n+11
n=3
2 times by 3=6+7=13
4 times by 3=12+11=23
So 4n+11 is larger than 2n+7
2n+7=4n+11
n=4
2 times by 4=8+7=15
4 times by 4=16+11=25
So 4n+11 is larger than 2n+7
2n+7=4n+11
n=5
2 times by 5=10+7=17
4 times by 5=20+11=21
So 4n+11 is larger than 2n+7
2n+7=4n+11
n=6
2 times by 6=12+7=19
4 times by 6=24+11=35
So 4n+11 is larger than 2n+7
2n+7=4n+11
n=7
2 times by 7=14+7=21
4 times by 7=28+11=39
So 4n+11 is larger than 2n+7
2n+7=4n+11
n=8
2 times by 8=16+7=23
4 times by 8=32+11=43
So 4n+11 is larger than 2n+7
2n+7=4n+11
n=9
2 times by 9=18+7=25
4 times by 9=36+11=47
So 4n+11 is larger than 2n+7
2n+7=4n+11
n=10
2 times by 10=20+7=27
4 times by 10=40+11=51
So 4n+11 is larger than 2n+7
The theory suggests that while both expressions rise as n increases,2(3n+4)’s total increases by 6 every interval,while 3(2n+4) total increases by the rate of 6 as well. The rate of increase may favor one expression over another beyond certain thresholds. Evaluating multiple values of n confirms this behavior and reinforces an understanding of linear functions.
2(3n+4)=3(2n+4)
n=1
2(3n+4)= 3 times by 1=3+4=7 times by 2=14
3(2n+4)=2 times by 1=2+4=6 times by 3=18
So 3(2n+4) is larger than 2(3n+4).
2(3n+4)=3(2n+4)
n=2
2(3n+4)=3 times by 2=6+4=10 times by 2=20
3(2n+4)=2 times by 2=4+4=8 times by 3=24
So 2(3n+43) is larger than 3(2n+4).
2(3n+4)=3(2n+4)
n=3
2(3n+4)=3 times by 3=9+4=13 times by 2=26
3(2n+4)=2 times by 3=6+4=10 times by 3=30
So 3(2n+4) is larger than 2(3n+4).
2(3n+4)=3(2n+4)
n=4
2(3n+4)=3 times by 4=12+4=16 times by 2=32
3(2n+4)=2 times by 4=8+4=12 times by 3=36
So 3(2n+4) is larger than 2(3n+4).
2(3n+4)=3(2n+4)
n=5
2(3n+4)=3 times by 5=15+4=19 times by 2=38
3(2n+4)=2 times by 5=10+4=14 times by 3=42
So 3(2n+4) is larger than 2(3n+4).
2(3n+4)=3(2n+4)
n=6
2(3n+4)=3 times by 6=18+4=22 times by 2=44
3(2n+4)=2 times by 6=12+4=16 times by 3=48
So 3(2n+4) is larger than 2(3n+4).
2(3n+4)=3(2n+4)
n=7
2(3n+4)=3 times by 7=21+4=25 times by 2=50
3(2n+4)=2 times by 7=14+4=18 times by 3=54
So 3(2n+4) is larger than 2(3n+4).
2(3n+4)=3(2n+4)
n=8
2(3n+4)=3 times by 8=24+4=28 times by 2=56
3(2n+4)=2 times by 8=16+4=20 times by 3=60
So 3(2n+4) is larger than 2(3n+4)
2(3n+4)=3(2n+4)
n=9
2(3n+4)=3 times by 9=27+4=31 times by 2=62
3(2n+4)=2 times by 9=18+4=22 times by 3=66
So 3(2n+4) is larger than 2(3n+4)
2(3n+4)=3(2n+4)
n=10
2(3n+4)=3 times by 10=30+4=34 times by 2=68
3(2n+4)=2 times by 10=20+4=24 times by 3=72
So 3(2n+4) is larger than 2(3n+4)
In this equation whatever n is both the formulas will be the same.
2(3n+3)=3(2n+2)
n=1
2(3n+3)=3 times by 1=3+3=6 times by 2=12
3(2n+2)=2 times by 1=2+2=4 times by 3=12
So 2(3n+3) and 3(2n+2) have the same value.
2(3n+3)=3(2n+2)
n=2
2(3n+3)=3 times by 2=6+3=9 times by 2=18
3(2n+2)=2 times by 2=4+2=6 times by 3=18
So 2(3n+3) and 3(2n+2) have the same value
2(3n+3)=3(2n+2)
n=3
2(3n+3)=3 times by 3=9+3=12 times by 2=24
3(2n+2)=2 times by 3=6+2=8 times by 3=24
So 2(3n+3) and 3(2n+2) have the same value
2(3n+3)=3(2n+2)
n=4
2(3n+3)=3 times by 4=12+3=15 times by 2=30
3(2n+2)=2 times by 4=8+2=10 times by 3=30
So 2(3n+3) and 3(2n+2) have the same value