SOLUTION: what are the answers for the following: -4<3x+1<=10 -2<=3x+8<-17 9>6-a -6x+3<=-32

Question 30023: what are the answers for the following:
-4<3x+1<=10
-2<=3x+8<-17
9>6-a
-6x+3<=-32
Answer by sdmmadam@yahoo.com(530) (Show Source): You can put this solution on YOUR website!
1)-4<3x+1<=10 ----(1)
1)-4<3x+1
-4-1 < 3x (grouping like terms)
-5 < 3x
Dividing by 3 and division by a positive quantity does not alter the inequality and so here less than remains less than
(-5/3) < x ----(*)
3x+1<=10
3x <= 10-1
3x <= 9
Dividing by 3
x <= 3 ----(**)
From (*) and (**) we have
(-5/3) < x <= 3
2)-2<=3x+8<-17 ----(2)
-2<=3x+8
-2-8 <= 3x
-10<= 3x
Dividing by 3 and division by a positive quantity does not alter the inequality and so here less than remains less than
(-10/3) <= x ----(1)
3x+8<-17
3x < -17-8
Dividing by 3
x < -25/3 ----(2)
But by (1), x is to the right of (-10/3) and that which is to the right of
(-10/3) CANNOT be to the left of (-25/3)
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.
So no solution.
That is the answer set is the empty set.
3)9>6-a
a > 6-9 (grouping like terms,changing sign while changing side)
a > -3
4)-6x+3<=-32
-6x <= -32-3
-6x <= -35
x >= (-35)/(-6)
Dividing by (-6) and division by a negative quantity
alters the inequality and so here less than becomes greater than.
That is x >= 35/6

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