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# 4v+8>6v+10

The short answer is that you solve it the same way as you would equations with one major exception: whenever you multiply or divide both sides with a negative number, you switch the inequality signs direction.

4v+8>6v+10
let's say you decide to collect the v terms on the left and the numbers on the right. Then subtracting 6v from both sides and subtracting 8 from both sides gives you

-2v > 2

This should be familiar. But now you need to divide by -2 to solve for v. Since this is a negative number you switch the freater than to a less than sign and get

v < -1

Had you decided insteal to collect v's on the right and numbers on the left, you would have subtracted 4v and 10 from both sides to get

-2 > 2v

Division by a positive number doesn't require the switch, so you wind up with the same result

-1 > v

Hi, Starr.  We are neighbors.  Please look at my profile to determine if you would like assistance in person from me.

The math problem you shared has one variable, v, located on two sides of the inequality.  To solve, it is important to combine like terms.  Below, I will work a similar problem.

7w + 3 < 2w - 5

I will subtract 2w from both sides; it's essential that whatever you do to one side, you do to the other side.  This means I will work "7w - 2w = 5w" on the left side.  On the right side, "2w - 2w = 0."  I chose to subtract the "2w" from both sides because subtraction is the inverse operation of addition.  On the right hand side of the inequality the "2w" originally was being combined (or added) to the -5.

Once the 2w is subtracted from both sides, the updated inequality if "5w +3 < - 5."  Now I need to combine the values that do NOT have a variable.  The goal is to leave the "5w" alone on the left side of the inequality.  So, I will subtract "3" from both sides of the inequality.  "+3 - 3 = 0" and "-5 - 3 = - 8."  (If you need a review about adding and subtracting positive and negative numbers, please let me know.)

At this point, we have the updated inequality "5w < -8,"  five time w is less than -8.  So, I use division, the inverse operation of multiplication.  5w divided by 5 = w.  -8 divided by 5 is -8/5.  This can also be re-written as -1 3/5 or -1.6.  (If you need a review about converting between improper fractions, mixed numbers, and decimals, please let me know.)  So, the final answer is w < -1.6.

This is a linear inequality, and the easiest way to solve it is to hypothetically start by turning it into an equation. For example in this case, instead of ">" let's use "="

Then, 4v + 8 = 6v + 10. To solve this, we group the like terms together on one side of the "=", such as 4v-6v=10-8, and when we solve for this, v=-1. This means that our original inequality would give an equality for v=-1.

Now, we need to see if our inequality would be true for values of v smaller and larger than -1.

Lets take a number smaller than -1 and plug it into our iniatial inequality.

Lets say v=-3 (smaller than -1). When replacing v with -3 in the inequality, we get: 4(-3)+8>6(-3)+10

-12+8>-18+10

-4>-8 This is a true expression, so we conclude that the inequality is true for all the values of v < -1.

Let's see what happens whith values larger than -1. Let's take v=1 (larger than -1). When we replace v with 1 in the inequality, we get: 4*1+8>6*1+10

4+8>6+10

12>16 This is a false expression, which means for all the v values larger than -1, the inequality is not true.

The answer then is v < -1.

ps. There are alternate ways of solving this, I thought this would be the easiest way to think of solving this logically. I hope it helps.