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1)-4<3x+1<=10 ----(1)
\n" ); document.write( "1)-4<3x+1
\n" ); document.write( "-4-1 < 3x (grouping like terms)
\n" ); document.write( "-5 < 3x
\n" ); document.write( "Dividing by 3 and division by a positive quantity does not alter the inequality and so here less than remains less than
\n" ); document.write( "(-5/3) < x ----(*)
\n" ); document.write( "3x+1<=10
\n" ); document.write( "3x <= 10-1
\n" ); document.write( "3x <= 9
\n" ); document.write( "Dividing by 3
\n" ); document.write( "x <= 3 ----(**)
\n" ); document.write( "From (*) and (**) we have
\n" ); document.write( "(-5/3) < x <= 3 \r
\n" ); document.write( "\n" ); document.write( "2)-2<=3x+8<-17 ----(2)
\n" ); document.write( "-2<=3x+8
\n" ); document.write( "-2-8 <= 3x
\n" ); document.write( "-10<= 3x
\n" ); document.write( "Dividing by 3 and division by a positive quantity does not alter the inequality and so here less than remains less than
\n" ); document.write( "(-10/3) <= x ----(1)
\n" ); document.write( "3x+8<-17
\n" ); document.write( "3x < -17-8
\n" ); document.write( "Dividing by 3
\n" ); document.write( "x < -25/3 ----(2)
\n" ); document.write( "But by (1), x is to the right of (-10/3) and that which is to the right of
\n" ); document.write( "(-10/3) CANNOT be to the left of (-25/3)
\n" ); document.write( "Please draw a horizontal line and mark O on it to represent the number zero. This line is your number line. Go 3 and (1/3) units to the left of O and mark A on this line. Go still left and mark B on this line such that OB =8 and (1/3) that is B is 5 units to the left of A. Now you can see,feel and understand how the two possibilities for x are contradictory.
\n" ); document.write( "So no solution.
\n" ); document.write( "That is the answer set is the empty set.\r
\n" ); document.write( "\n" ); document.write( "3)9>6-a
\n" ); document.write( "a > 6-9 (grouping like terms,changing sign while changing side)
\n" ); document.write( "a > -3\r
\n" ); document.write( "\n" ); document.write( "4)-6x+3<=-32
\n" ); document.write( "-6x <= -32-3
\n" ); document.write( "-6x <= -35
\n" ); document.write( "x >= (-35)/(-6)
\n" ); document.write( "Dividing by (-6) and division by a negative quantity
\n" ); document.write( "alters the inequality and so here less than becomes greater than.
\n" ); document.write( "That is x >= 35/6\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "