Question

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

Answer 1

-6`>=`y

Explanation:

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

Answer 2

`y<=-6`

Explanation:

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`.