What will our answer look like? A bunch of numbers, if past experience is anything to go by. We do have a "greater than" sign, though, so it will also be an "or" inequality. Let's look at the first part of the inequality, when everything looks the same except we drop the absolute value bars:
15 – 2x ≥ 17
-2x ≥ 2
Danger, danger. Swap that sign when dividing by a negative.
x ≤ -1
Now for the other set. We reverse the polarity outside the absolute value bars, and we reverse the sign of the inequality.
15 – 2x ≤ -17
-2x ≤ -32
We've flipped so many signs, you could call us Vanna White.
x ≥ 16
Our complete set of solutions is x ≤ -1 or x ≥ 16.
Example 2
Solve |4x + 5| < 21.
Our inequality has a "less than" sign. It means we have a compound "and" inequality, with the results between +21 and -21. The setup for it is:
-21 < 4x + 5 < 21
From here, it's a standard treasure hunt for x.
-26 < 4x < 16
We couldn't even get a cup of coffee with this. That isn't much of a treasure at all.
Example 3
Solve .
Well, that's a sticky wicket. We better simplify this, stat.
|8x + 4 – 2 + 2x| < 9
|10x + 2| < 9
Much nicer. Let's finish off this by finding its compound "and" solutions.
.