"Hard" Math Questions. Help Please?

Question

-Jack wrote 100 positive numbers between 0 and 3, but he did not write the number 1. Jill wrote the reciprocal of every one of Jack's numbers. How many of these 200 numbers were between 0 and 1?

-In a country whose currency consists of $6 bills and $11 bills, the price of every item sold is a whole number of dollars that can be paid exactly, without need for "change." using only these two types of bills. What is the largest whole number of dollars which could NOT be the price of an item sold in this country?

-What are all real values of x for which (absolute value) x^2-4(end absolute value) -3x<(or equal to) 0?

Detailed explanations please!

Thanks.

Answer 1

1)  Half of them.  If Jack never used the number 1, then his numbers must all have been 0 &lt; x &lt; 1 and 1 &lt; x &lt; 3.  Any number he chose from the first group would have a reciprocal greater than 1.  So, if he chose 25 numbers from the first group, Jill would have 25 numbers greater than 1, so that would give us the same number greater than 1 and between 0 and 1.  Any numbers Jack chose from the second group would have a reciprocal between 0 and 1.  So, if he chose 75 numbers in the second group, Jill would have 75 numbers between 0 and 1.  However you work it, if Jack&#039;s number was between 0 and 1, Jill&#039;s wasn&#039;t.  If Jack&#039;s number was not between 0 and 1, Jill&#039;s number would be.  They&#039;ll balance each other out.

I apologize, but the other two require more thought than I&#039;m willing to put into this right now.  Hope the first one helps.

Answer 2

for the 3rd question:

I x^2-4 I  * 3x &lt;(or equal to) 0

x^2-4 * 3x &lt;(or equal to) 0         x^2-4 * -3x &gt;(or equal to) 0

x^2 * 3x - 4 &lt;(or equal to) 0       x^2 * -3x - 4 &gt;(or equal to) 0

[now factor or use quadratic equation]

(x+4)(x-1) &lt;(or equal to) 0         (x-4)(x+1) &gt;(or equal to) 0

[then find the values of the x&#039;s]

x &lt;(or equal to) -4 &amp; 1                  x &gt;(or equal to) 4 &amp; -1

so answer would be on # line:

                -5  -4  -3  -2  -1  0  1  2  3  4  5

     &lt;---------------------------------------...

                      &lt;-------------------------------&gt;                

(u should color that arrow line that represents &#039;x&#039; on the main # line [ color over the original # line)

sorry if that isnt rite :(   but i think it is :)

Answer 3

100 (think about it!) if a number is &gt; 1 than its reciprocal is &lt; 1

any amount (in a whole number of dollars) because 1 = 5*11 - 9*6

1 &lt;= x &lt;= 4  (draw the graphs)