"Solve" is just a word for "get x all alone and see what it fesses up to." Kick off this interrogation by subtracting 3 from both sides.
-6x < 12
Our Shmoopy sense is tingling. We divide by -6 with the utmost caution. Don't forget to flip that inequality sign around.
x > -2
Our answer says that the original inequality will be true any time we pick an x larger than -2. So how about it? Let's pick something and check it out. We could go for something really big (we're partial to 1490.723), but even something as simple as x = 0 will do.
-6(0) + 3 < 15
3 < 15
Yep, that's a true statement, so our answer is right. We got the confession we were after.
Example 2
Solve 2y – 1 ≥ 4y + 2.
Looks like x is on its lunch break, so y is covering its shift. We'll just go through things as though x were still here.
-3 ≥ 2y
We decided to add things together so that y could stay positive. It's here to help out a friend; there's no need to be a negative jerk. However, we like having our variable on the left side, so we'll rewrite the equation. Watch the inequality sign carefully:
2y ≤ -3
It was pointing away from y when we started, and it still is. That's how we know we moved everything correctly.
We avoided negative division, so the problem ends with not with a bang, but with the inequality sign staying in place.
Example 3
Solve .
Yuck, a fraction on our variable. Kill it with fire, stomp it flat, multiply it out: we don't care how, just get rid of it.
Uhohohoh, look out. We have multiplication by a number less than 0. Not only do we have to change every terms' sign, but also the direction of the inequality.
x – 7 ≥ -2x – 10
Now we can finish solving for x.
3x ≥ -3
x ≥ -1
We'll check our answer by plugging in x = 1, which gets us 3 ≤ 6. That's a winner.
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