71 A student asks When we write a S b we mean that a is less

71. A student asks, \"When we write a S b, we mean that a is less than b or a is equal to b. But when we write a b, you say that this means that a is not less than b and a is not equal to b. I don\'t understand why it changes from or to and.\" 72. While studying a unit on inequalities, a student asks you if it\'s true that if a b and c > d 73. In your ninth-grade algebra class, you are asked, \"Does (a + b)(c + d) = ac + bd?\" 74. In a discussion on rates, a student says, \"If I walk 4 mph for one mile, \'ll have to run 8 mph 75, A student asks you if a+b = a + b. oll then a c>b-d. the second mile in order to average 6 mph for 2 iles. Right?\"

Solution

Solution:

71. when we write ab in this case we can have two condition like either a<b or a=b and if anyone of them is ture that means this inequality is true.

But if we write “a“ is not less than ”b“ and ab that means neither of both is true therefore both of them must be true to hold that inequality ture (not less than and )

hence we use \"and\" for the last inequality

72.

If a>b and c>d then a-c > b-d

this condition is not true

we can prove it by simple example

let a = 5 and b = 4 here 5 > 4 hence a>b

and c = 3 and d = 1 and here c>d becuase 3>1

a-c = 2 and b-d = 3

thus 2>3 is not a valid inequality therefore a-c > b-d is not ture

(73) (a +b ) (c+d) ac + bd

this again is not ture

because we use FOIL method to expand this multiplication

(a +b ) (c + d) = ac + ad + bc +bd

answer

(74)

we know that average speed = total distance / total time taken to cover the distance

distance travelled with 4mph = 1 mile and time taken is = 1/4 ( time = distance / speed )

distance travelled with 8mph = 1 mile and time taken is = 1/8 (time = distance / speed )

therefore average speed = total distance ( 1+1 ) / total time taken ( 1/4 +1/8)

= 2 / (1/4+1/8)

= 2/ (3/8)

= 16/3 mph

hence averge 6 mph for 2 mph is not correct

(75)

(1 / (a +b)) = 1/a + 1/b

this is not true

because 1/a +1/b = (a+b)/ab

which is not equal 1/(a+b)

we can understant more with example

let a = 1 and b = 2

1 /(a+b) = 1/(1+2) = 1/3 ...1

and 1/a + 1/b = 1/1 +1/2= 3/2 ...2

from 1 and 2

1/(a+b) 1/a +1/b

hence proved