How do you solve -17+ 3y + 7y \geq 19+ 16y17+3y+7y19+16y?

2 Answers
May 18, 2018

-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

May 18, 2018

y<=-6y6

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< ba<b we mean to say that aa is closer to 00 than bb. But, if we consider -aa and -bb, we will notice that -a < -ba<b is false because -aa is closer to 00 than -bb. 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+16y17+3y+7y19+16y.

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

-17+3y+7y>=19+16y = -17+10y>=19+16y17+3y+7y19+16y=17+10y19+16y.

Adding 1717 to both sides, we obtain

10y>=36+16y10y36+16y.

Now we subtract 16y16y from both sides:

-6y>=366y36.

Now, to further simplify, we must divide by -66, and we can, but we must also remember to flip the inequality when we do so. We obtain:

y<=-6y6.