Now examine the sum of a rational number, y, and an irrational number, x. The rational number y can be written as y = a/b, where a and b are integers and b ≠ 0. Leave the irrational number x as x because it can’t be written as the ratio of two integers. Let’s look at a proof by contradiction. In other words, we’re trying to show that x + y is equal to a rational number instead of an irrational number. Let the sum equal m/n, where m and n are integers and n ≠ 0. The process for rewriting the sum for x is shown.

Statement l Reason

x + a/b = m/n l substitution

x + a/b - a/b = m/n - a/b l subtraction property of equality

x = m/n -a/b l

x = (b/b) (m/n) -( n/n) (a/b) l Create common denominators.
x = bm/bn - an/bn l

x = bm-an/bn l Simplify

Based on what we established about the classification of x and using the closure of integers, what does the equation tell you about the type of number x must be for the sum to be rational? What conclusion can you now make about the result of adding a rational and an irrational number?

step-by-step explanation:

always plug in x. in this case, x equals 4.

i'm going to first solve 4(2+3x)

1) 4(2+3(4)) use pemdas to solve.

i'm going to multiply whats in the parenthesis first, so 3 times 4=12.

2)4 (2+12) i'm then going to multiply 4 by both numbers in the parenthesis (because multiplying comes before adding)

4 times 2= 8.   4 times 12= 48

4) (8+48)= 56 (answer)

i'm then going to solve 8+12x and plug in 4 for x.

1) 8 +12(4)

multiplying comes before adding, so i'm gonna multiply 12 and 4.

12 times 4= 48.

2) 8+48=56

both equal the same!

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.

- both graphs increase on their domain.

- the graphs of both functions have the same x-intercept.

- neither graph has a y-intercept.