Question

Which number is a solution or the inequality 12<y(8-y)

a.0
b.1
c.2
d.3

if n is the set of natural numbers that are factors of 20 choose the selection below that correctly shows this set in roster form
{2,4,5,10}
{2,4,6,8,10,12,14,16,18,20}
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
{1,2,4,5,10,20}

I will give the person 100 points for whoever gives me the right answer!
(no links or I'll report you.)​

Answer 1

see below

Explanation:

1) given inequality,

12 < y (8-y)

simplify,

12 < 8y - y²

can be written as,

8y - y² > 12

8y - y² -12 > 0

factor out the LHS as ,

-1( -8y + y² +12) > 0

-1( y² - 6y - 2y + 12 ) > 0

-1{ y ( y -6) -2(y -6) } > 0

-1 (y -6)( y -2) > 0

( 6 -y ) (y -2) > 0

6 - y > 0 , y - 2 > 0

y < 6 , y > 2

so we have 2 < y < 6

a value between 2 and 6 is 3 .

option d is correct .

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2 ) all factors of 20 are ,

1 , 2 , 4 , 5 , 10 , 20

in roster form ,

{1,2,4,5,10,20}

last option is correct.

and we are done!

Answer 2

The number that is a solution of the given inequality is 3.

The set in roster form is: {1, 2, 4, 5, 10, 20}.

Explanation:

To solve the given inequality, first rearrange the inequality using algebraic operations so that all terms are on one side:

`\implies 12 < y(8-y)`

`\implies 12 < 8y-y^2`

`\implies y^2-8y+12 < 0`

Factor the left side of the inequality:

`\implies y^2-2y-6y+12 < 0`

`\implies y(y-2)-6(y-2) < 0`

`\implies (y-6)(y-2) < 0`

Therefore, the solution to the inequality is:

• 
  `2 < y < 6`

Therefore, the number that is a solution from the given answer options is 3.

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Natural numbers are positive integers that start from 1.

A factor is an integer that divides exactly into a whole number without a remainder.

In roster form, all the elements of a set are listed (separated by commas) and enclosed within braces { }.

The factors of 20 that are natural numbers are:

• 1, 2, 4, 5, 10 and 20.

Therefore, if n is the set of natural numbers that are factors of 20, the set shown in roster form is:

• {1, 2, 4, 5, 10, 20}