4 < -2x + 4 < 12
With one exception (which we'll get to), inequalities work exactly like equations.
If you had:
-2x + 4 = 12
you would have no problem solving that.
First, you would subtract 4 from both sides:
-2x = 8
Then, you would divide both sides by -2:
x = -4
No problem!
You're doing almost the same thing with inequalities. If you had a single inequality, like
-2x + 4 < 12
you would subtract 4 from both sides, same as with an equation:
-2x < 8
NOW is the only exception that comes with inequalities! WHEN YOU MULTIPLY OR DIVIDE BY A NEGATIVE, YOU MUST SWITCH THE INEQUALITY SIGN. Until now, we've just left "less-than" as "less-than". Now, when we divide both sides by -2, we must change the "less-than" to a "greater-than":
x > -4
That's the only trick!
Now, with your equation, you have THREE SIDES. That might be throwing you off a bit, but don't let it! Before, you did everything to BOTH sides of an equation or inequality. Now, you do everything to ALL THREE sides. No problem!
So, you have:
-4 < -2x + 4 < 12
Start by subtracting 4, same as before. Only now you do it to all three sides:
-8 < -2x < 8
Next, you have to get rid of that -2, so you divide ALL THREE sides by -2. DON'T FORGET TO SWITCH THE SIGNS:
4 > x > -4
And you're done!
4 > x > -4
It sometimes helps to rewrite a three-part inequality as two seperate problems:
-4 < -2x + 4
-2x + 4 < 12
Don't hesitate to do this, especially for larger, messier problems! You'll get exactly the same answers!