Question 27881
Graphs/27881 (2006-02-20 01:56:31): Tell what you must do the first inequality in order to get second. Be sure to list all the steps.show me how to Solve it 
1.)4j+5>=23;j>=4.5 

4j+5>=23
4j>=(23-5)   (subtracting 5 from both the sides)
4j>=18
j>=18/4  (dividing by 4>0, multip or dividing by a positive quantity  does not alter the inequality sign)
j>=(2X9)/(2X2) = 9/2 = 4.5
Therefore j>=4.5 

2.) 2(q-3)<8;q<7 
2(q-3)<8
2q-6<8
2q<8+6  (adding 6 to both the sides of the inequality)
2q<14
q<14/2 
That is  q <7


3.) 8-4s>16;-4s>8
8-4s>16
8-4s+4s> 16+4s  (adding 4s to both the sides) 
8+0 > 16+4s
8>16+4s
8-16 > +16+4s -16 (subtracting 16 from both the sides)
-8> 4s+16-16  (using additive commutativity on the RHS)
-8>4s+0
-8>4s
4s <-8  (Adam older than Tom is the same as Tom younger to Adam)
Dividing by 4>0 
(and multiplying or dividing an inequality by a postitive quantity keeps the  inequality unaltered.)
s<(-2)
 

4.) -8>z/(-5)-2
-8>(z/-5)-2
-8+2 >(z/-5) -2+2  (adding 2 to both the sides)
-6>z/(-5) +0
-6>z/(-5)
Multiplying both the sides by (-5)
(-6)X(-5) < z 
(why change in the inequality? 
It is because mulltiplication or division by a negative quantity  alters the inequality)
That  is   30 <z
or  z>30



5.) 2y-5>9+y;y>14 
2y-5>9+y
2y-5-y >9+y-y  (subtracting y from both the sides or adding -y to both the sides)
2y-y-5> 9+0  (using additive associativity,commutativity on the LHS)
y-5>9
y -5+5 >9+5  (adding  5 to both the sides)
y+0 >14
y>14


6.) 2/3g+7>=9;2/3g>=2] 
2/3g+7>=9
2/3g+7-7 >=9-7 (adding (-7) to both the sides)
2/3g+0 >=2
2/3g >=2
[g>=2X3/2  (multiplying by 3/2>0)
g>=3]


7.) 6<12-s;s<6 

6<12-s
6+s<12-s+s  (adding s to both the sides)
6+s<12+0
6+s<12
-6+6+s< -6+12  (adding (-6) to both the sides)
0+s <6
s<6

8.) 3+5t>=6(t-1)-t;3>=-6    
Is the problem  1)3+5t>=6(t-1)  OR  2)3+5t>=6(t-1)-t

1)   	3+5t>=6(t-1)
	3+5t>=6t-6
	3+6 >=6t-5t   (you can simultaneously add 6 to  both the sides and subtract 5t from both the sides, 		in fact it is like things being transfered from one side to another , changing sign  while changing side)
	9>=t 
2)	3+5t>=6(t-1)-t
	3+5t>=6t-6-t
	3+5t>=6t-t-6
	3+5t>=5t-6
	3+5t-5t>= 5t-6-5t
	3+0>=-6+5t-5t
	3>=-6+0
	3>=-6
	Note as 3 cannot be equal to -6
	we have 3>-6


9.) 6.2<-r;-6.2>r 

6.2<-r
Multiplying by (-1) on both the sides, we have
-6.2 > (-1)X(-r)   [multiplication by a negative quantity alters the inequality]
-6.2>+r
-6.2>r
Solve it and graph it 
I do not know todo the technique of doing graphs on your answer sheet. 
My graphs are not getting copied to the answer sheet.

10.) 2x-2>4
Dividing by 2>0  (dividing by a positive quantity does not alter the inequality)
x-1>2
x-1+1>2+1 (adding 1 on both the sides)
x+0>3
x>3 

11.) 2-2x>4 
Dividing by 2>0  (dividing by a positive quantity does not alter the inequality)
1-x>2
-2+(1-x)+x>-2+(2)+x 
(as you need to  transfer 2 from the right to left   and (-x ) from the left  to right,we add-2 from the left on both the sides and +x from the right on both the sides)
(-2+1)+(-x+x) >[-2+2] +x   (by additive associativity on both the sides)
-1+0>0+x
-1>x
x<-1
Note: If you are asked to every step with this particular problem in focus, then do as above. 
But otherwise if such an inequality is a portion of a bigger problem  then you may do it  quickly as  follows:
2-2x>4 
1-x>2 (Dividing by2>0)
1-2>+x  
(you may transfer things  from one side to another changing sign for the quantity 
everytime while changing side)
-1>x
x<-1   (using a>b implying  b<a)

12.) 2x+2>4
x+1>2  (dividing by 2)
x>2-1
x>1

13.) 2x+2>4x 
x+1>2x  (dividing by 2)
1>2x-x
1>x
x<1
14.) -2x-2>4 
  -x-1>2
-x>2+1
-x>3
x<-3  (Multiplying by (-1) and multiplication by a negative quantity alters the inequality)

15.) -2(x-2)>4 
-1(x-2)>2
-x+2>2
-x>2-2
-x>0
(-1)X (-x)<(-1)X0  (multiplying by (-1) and multiplication by a negative quantity alters the inequality)
x<0

Solve each inequality. Graph the solutions on a number line. 
17.) 5<=11+3h 
5-11<=3h
-6<=3h
-2<=h  (dividing by 3>0)
h>=-2   (using a<=b means b>=a)

18.) 3(y-5)>6 
(y-5)>2  (dividing by3>0)
y>2+5
y>7

19.) -4x-2<8 
  -2x-1<4  (dividing by 2>0)
 -2x<4+1
-2x<5
x>5/(-2)    (dividing by (-2) and dividing by a negative  quantity alters the inequality)
x>(-5/2)

20.) r+6+3r>=15-2r
       r +3r +2r >=15-6 
(transferring (-2r) from the right to the left  and transferring 6 from  the left to the right)
         6r>=9
          r>=9/6
          r>=3/2

21.) 5-2n<=3-n 
      5-3<=2n-n  (transferring 3 from the right to the left and (-2n) from the left to the right )
       2<=n
That is  n>=2

22.) 3(2v-4)<=2(3v-6)
    6v-12<=6v-12
Note: (6v-12) cannot be less than itself
Therefore 6v-12 = 6v-12 alone holds  and this is true for all values of v 
23.) 2(m-8)-3m<-8
  2m-16 -3m<-8
2m-3m-16<-8  (additive commutativity and associativity)
-m<-8+16
-m<8
m>-8   (multiplying by (-1) multiplication by a negative  quantity alters the inequality)

24.) -(6b-2)>0
    -6b +2  >0
     2>6b   (transferring (-6b) from the left to the right )
     Dividing by 6
     2/6>b
     1/3>b
  That  is b<1/3


25.) 7a-(9a+1)>5 
    7a-9a-1>5
     -2a>5+1
  -2a>6
  a<6/(-2)  ( dividing by (-2) amd dividing by a negative quantity alters the inequality)
  a<-3