How do you solve #-17+ 3y + 7y \geq 19+ 16y#?

Collect the like terms on the left hand side
-17+10y#>=#19+16y
Take 10y from each side so that you only have y on 1 side
-17#>=#19+6y
Take 19 from each side
-36#>=#6y
Finally divide each side by 6
-6#>=#y

Solving an inequality is almost exactly like solving an equality, and for the most part you can treat it as such while solving it, except for one additional rule: whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, #># would go to #<#, #<=# to #>=# and vice versa. If you wish to know why you must do this, read the next paragraph; otherwise, you may skip it.

The reason this rule arises is because of how the number line works. Observe that if we write #a< b# we mean to say that #a# is closer to #0# than #b#. But, if we consider #-a# and #-b#, we will notice that #-a < -b# is false because #-a# is closer to #0# than #-b#. Therefore, when we manipulate inequalities by multiplying or dividing by a negative we must flip the inequality symbol to accurately reflect which expression is closer to zero.

Now we will solve the inequality

#-17+3y+7y>=19+16y#.

So, to begin, we can solve this inequality exactly like solving an equality:

#-17+3y+7y>=19+16y = -17+10y>=19+16y#.

Adding #17# to both sides, we obtain

# 10y>=36+16y#.

Now we subtract #16y# from both sides:

# -6y>=36#.

Now, to further simplify, we must divide by #-6#, and we can, but we must also remember to flip the inequality when we do so. We obtain:

#y<=-6#.