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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
\n" ); document.write( "1.)4j+5>=23;j>=4.5 \r
\n" ); document.write( "\n" ); document.write( "4j+5>=23
\n" ); document.write( "4j>=(23-5) (subtracting 5 from both the sides)
\n" ); document.write( "4j>=18
\n" ); document.write( "j>=18/4 (dividing by 4>0, multip or dividing by a positive quantity does not alter the inequality sign)
\n" ); document.write( "j>=(2X9)/(2X2) = 9/2 = 4.5
\n" ); document.write( "Therefore j>=4.5 \r
\n" ); document.write( "\n" ); document.write( "2.) 2(q-3)<8;q<7
\n" ); document.write( "2(q-3)<8
\n" ); document.write( "2q-6<8
\n" ); document.write( "2q<8+6 (adding 6 to both the sides of the inequality)
\n" ); document.write( "2q<14
\n" ); document.write( "q<14/2
\n" ); document.write( "That is q <7\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "3.) 8-4s>16;-4s>8
\n" ); document.write( "8-4s>16
\n" ); document.write( "8-4s+4s> 16+4s (adding 4s to both the sides)
\n" ); document.write( "8+0 > 16+4s
\n" ); document.write( "8>16+4s
\n" ); document.write( "8-16 > +16+4s -16 (subtracting 16 from both the sides)
\n" ); document.write( "-8> 4s+16-16 (using additive commutativity on the RHS)
\n" ); document.write( "-8>4s+0
\n" ); document.write( "-8>4s
\n" ); document.write( "4s <-8 (Adam older than Tom is the same as Tom younger to Adam)
\n" ); document.write( "Dividing by 4>0
\n" ); document.write( "(and multiplying or dividing an inequality by a postitive quantity keeps the inequality unaltered.)
\n" ); document.write( "s<(-2)
\n" ); document.write( " \r
\n" ); document.write( "\n" ); document.write( "4.) -8>z/(-5)-2
\n" ); document.write( "-8>(z/-5)-2
\n" ); document.write( "-8+2 >(z/-5) -2+2 (adding 2 to both the sides)
\n" ); document.write( "-6>z/(-5) +0
\n" ); document.write( "-6>z/(-5)
\n" ); document.write( "Multiplying both the sides by (-5)
\n" ); document.write( "(-6)X(-5) < z
\n" ); document.write( "(why change in the inequality?
\n" ); document.write( "It is because mulltiplication or division by a negative quantity alters the inequality)
\n" ); document.write( "That is 30
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\n" ); document.write( "\n" ); document.write( "5.) 2y-5>9+y;y>14
\n" ); document.write( "2y-5>9+y
\n" ); document.write( "2y-5-y >9+y-y (subtracting y from both the sides or adding -y to both the sides)
\n" ); document.write( "2y-y-5> 9+0 (using additive associativity,commutativity on the LHS)
\n" ); document.write( "y-5>9
\n" ); document.write( "y -5+5 >9+5 (adding 5 to both the sides)
\n" ); document.write( "y+0 >14
\n" ); document.write( "y>14\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "6.) 2/3g+7>=9;2/3g>=2]
\n" ); document.write( "2/3g+7>=9
\n" ); document.write( "2/3g+7-7 >=9-7 (adding (-7) to both the sides)
\n" ); document.write( "2/3g+0 >=2
\n" ); document.write( "2/3g >=2
\n" ); document.write( "[g>=2X3/2 (multiplying by 3/2>0)
\n" ); document.write( "g>=3]\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "7.) 6<12-s;s<6 \r
\n" ); document.write( "\n" ); document.write( "6<12-s
\n" ); document.write( "6+s<12-s+s (adding s to both the sides)
\n" ); document.write( "6+s<12+0
\n" ); document.write( "6+s<12
\n" ); document.write( "-6+6+s< -6+12 (adding (-6) to both the sides)
\n" ); document.write( "0+s <6
\n" ); document.write( "s<6\r
\n" ); document.write( "\n" ); document.write( "8.) 3+5t>=6(t-1)-t;3>=-6
\n" ); document.write( "Is the problem 1)3+5t>=6(t-1) OR 2)3+5t>=6(t-1)-t\r
\n" ); document.write( "\n" ); document.write( "1) 3+5t>=6(t-1)
\n" ); document.write( " 3+5t>=6t-6
\n" ); document.write( " 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)
\n" ); document.write( " 9>=t
\n" ); document.write( "2) 3+5t>=6(t-1)-t
\n" ); document.write( " 3+5t>=6t-6-t
\n" ); document.write( " 3+5t>=6t-t-6
\n" ); document.write( " 3+5t>=5t-6
\n" ); document.write( " 3+5t-5t>= 5t-6-5t
\n" ); document.write( " 3+0>=-6+5t-5t
\n" ); document.write( " 3>=-6+0
\n" ); document.write( " 3>=-6
\n" ); document.write( " Note as 3 cannot be equal to -6
\n" ); document.write( " we have 3>-6\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "9.) 6.2<-r;-6.2>r \r
\n" ); document.write( "\n" ); document.write( "6.2<-r
\n" ); document.write( "Multiplying by (-1) on both the sides, we have
\n" ); document.write( "-6.2 > (-1)X(-r) [multiplication by a negative quantity alters the inequality]
\n" ); document.write( "-6.2>+r
\n" ); document.write( "-6.2>r
\n" ); document.write( "Solve it and graph it
\n" ); document.write( "I do not know todo the technique of doing graphs on your answer sheet.
\n" ); document.write( "My graphs are not getting copied to the answer sheet.\r
\n" ); document.write( "\n" ); document.write( "10.) 2x-2>4
\n" ); document.write( "Dividing by 2>0 (dividing by a positive quantity does not alter the inequality)
\n" ); document.write( "x-1>2
\n" ); document.write( "x-1+1>2+1 (adding 1 on both the sides)
\n" ); document.write( "x+0>3
\n" ); document.write( "x>3 \r
\n" ); document.write( "\n" ); document.write( "11.) 2-2x>4
\n" ); document.write( "Dividing by 2>0 (dividing by a positive quantity does not alter the inequality)
\n" ); document.write( "1-x>2
\n" ); document.write( "-2+(1-x)+x>-2+(2)+x
\n" ); document.write( "(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)
\n" ); document.write( "(-2+1)+(-x+x) >[-2+2] +x (by additive associativity on both the sides)
\n" ); document.write( "-1+0>0+x
\n" ); document.write( "-1>x
\n" ); document.write( "x<-1
\n" ); document.write( "Note: If you are asked to every step with this particular problem in focus, then do as above.
\n" ); document.write( "But otherwise if such an inequality is a portion of a bigger problem then you may do it quickly as follows:
\n" ); document.write( "2-2x>4
\n" ); document.write( "1-x>2 (Dividing by2>0)
\n" ); document.write( "1-2>+x
\n" ); document.write( "(you may transfer things from one side to another changing sign for the quantity
\n" ); document.write( "everytime while changing side)
\n" ); document.write( "-1>x
\n" ); document.write( "x<-1 (using a>b implying b
\n" ); document.write( "x+1>2 (dividing by 2)
\n" ); document.write( "x>2-1
\n" ); document.write( "x>1\r
\n" ); document.write( "\n" ); document.write( "13.) 2x+2>4x
\n" ); document.write( "x+1>2x (dividing by 2)
\n" ); document.write( "1>2x-x
\n" ); document.write( "1>x
\n" ); document.write( "x<1
\n" ); document.write( "14.) -2x-2>4
\n" ); document.write( " -x-1>2
\n" ); document.write( "-x>2+1
\n" ); document.write( "-x>3
\n" ); document.write( "x<-3 (Multiplying by (-1) and multiplication by a negative quantity alters the inequality)\r
\n" ); document.write( "\n" ); document.write( "15.) -2(x-2)>4
\n" ); document.write( "-1(x-2)>2
\n" ); document.write( "-x+2>2
\n" ); document.write( "-x>2-2
\n" ); document.write( "-x>0
\n" ); document.write( "(-1)X (-x)<(-1)X0 (multiplying by (-1) and multiplication by a negative quantity alters the inequality)
\n" ); document.write( "x<0\r
\n" ); document.write( "\n" ); document.write( "Solve each inequality. Graph the solutions on a number line.
\n" ); document.write( "17.) 5<=11+3h
\n" ); document.write( "5-11<=3h
\n" ); document.write( "-6<=3h
\n" ); document.write( "-2<=h (dividing by 3>0)
\n" ); document.write( "h>=-2 (using a<=b means b>=a)\r
\n" ); document.write( "\n" ); document.write( "18.) 3(y-5)>6
\n" ); document.write( "(y-5)>2 (dividing by3>0)
\n" ); document.write( "y>2+5
\n" ); document.write( "y>7\r
\n" ); document.write( "\n" ); document.write( "19.) -4x-2<8
\n" ); document.write( " -2x-1<4 (dividing by 2>0)
\n" ); document.write( " -2x<4+1
\n" ); document.write( "-2x<5
\n" ); document.write( "x>5/(-2) (dividing by (-2) and dividing by a negative quantity alters the inequality)
\n" ); document.write( "x>(-5/2)\r
\n" ); document.write( "\n" ); document.write( "20.) r+6+3r>=15-2r
\n" ); document.write( " r +3r +2r >=15-6
\n" ); document.write( "(transferring (-2r) from the right to the left and transferring 6 from the left to the right)
\n" ); document.write( " 6r>=9
\n" ); document.write( " r>=9/6
\n" ); document.write( " r>=3/2\r
\n" ); document.write( "\n" ); document.write( "21.) 5-2n<=3-n
\n" ); document.write( " 5-3<=2n-n (transferring 3 from the right to the left and (-2n) from the left to the right )
\n" ); document.write( " 2<=n
\n" ); document.write( "That is n>=2\r
\n" ); document.write( "\n" ); document.write( "22.) 3(2v-4)<=2(3v-6)
\n" ); document.write( " 6v-12<=6v-12
\n" ); document.write( "Note: (6v-12) cannot be less than itself
\n" ); document.write( "Therefore 6v-12 = 6v-12 alone holds and this is true for all values of v
\n" ); document.write( "23.) 2(m-8)-3m<-8
\n" ); document.write( " 2m-16 -3m<-8
\n" ); document.write( "2m-3m-16<-8 (additive commutativity and associativity)
\n" ); document.write( "-m<-8+16
\n" ); document.write( "-m<8
\n" ); document.write( "m>-8 (multiplying by (-1) multiplication by a negative quantity alters the inequality)\r
\n" ); document.write( "\n" ); document.write( "24.) -(6b-2)>0
\n" ); document.write( " -6b +2 >0
\n" ); document.write( " 2>6b (transferring (-6b) from the left to the right )
\n" ); document.write( " Dividing by 6
\n" ); document.write( " 2/6>b
\n" ); document.write( " 1/3>b
\n" ); document.write( " That is b<1/3\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "25.) 7a-(9a+1)>5
\n" ); document.write( " 7a-9a-1>5
\n" ); document.write( " -2a>5+1
\n" ); document.write( " -2a>6
\n" ); document.write( " a<6/(-2) ( dividing by (-2) amd dividing by a negative quantity alters the inequality)
\n" ); document.write( " a<-3