Sum/difference:
Let
This means that
Now, assume that 
is rational. The sum/difference of two rational numbers is still rational (so 5-x is rational), and the division by 3 doesn't change this. So, you have that the square root of 8 equals a rational number, which is false. The mistake must have been supposing that was rational, which proves that the sum/difference of the two given terms was irrational
Multiplication/division:
The logic is actually the same: if we multiply the two terms we get
if again we assume x to be rational, we have
But if x is rational, so is -x/15, and again we come to a contradiction: we have the square root of 8 on one side, which is irrational, and -x/15 on the other, which is rational. So, again, x must have been irrational. You can prove the same claim for the division in a totally similar fashion.
Answer:
The team won 32 games.
Step-by-step explanation:
x = games won
y = games lost
z = total games = 40
x = 4y <em>Won 4 times as many games as it lost</em>
x + y = 40 <em>Games won + games lost = Total games played</em>
4y + y = 40 <em>Sub 4y in for x, as established previously</em>
5y = 40
y = 8
x = 4y
x = 4(8)
x = 32
The team won 32 games.
7,440 I believe, all you need to use is basic subtraction for this.
1/2 = 1/1/2 cups
1 cup = 1/3 tablespoons
i am a mathematics teacher. if anything to ask please pm me
