I was trying to solve an inequality with a square root (to practice for admission test), this is the inequality:
7x + 2 < sqrt(8x^2 + x - 9) + x
Here's how I solved it: 1st, I've moved the square root from the right side to the left side, to flip the sign of the inequality and the sign of each member:
sqrt(8x^2 + x - 9) > -x + 7x + 2
2nd, now I've split the solutions into two different cases:
first system of inequalities:
8x^2 + x - 9 >= 0
-x + 7x + 2 > 0
8x^2 + x - 9 > (-x + 7x + 2)^2
second system:
-x + 7x + 2 < 0
3rd, I've solved each inequalities:
first system of inequalities:
8x^2 + x - 9 >= 0 --> x < -9/8 and x < 1
-x + 7x + 2 > 0 --> x < -1/3
8x^2 + x - 9 > (-x + 7x + 2)^2 --> 36x^2 + 24x + 4 > 0 --> Doesn't have real solutions (delta is negative) but since the inequality has "greater sign", then I can say it accepts every real solutions.
second system: -x + 7x + 2 < 0 --> x > -1/3
4th, now I've to find the common solutions in the number line: my result is only the empty set of possible solutions.
This exercise should give "x <= -9/8".
if you know a quickest method to solve these kind of exercises, let me know because the day of the official test I'll have only 50 minutes.
EDIT (solved): it's possible to solve this kind of inequalities using another method.
- find the domain of the square root (in this case x<-9/8 or x>1).
- test by plugging arbitrary values inside the inequality (in my case, x = 2)
- Let's see what happens. (in my case, 2 isn't a solution because 16 isn't less than 5.
When you inspect the two cases, you are saying that the condition described in the problem is the logical disjunction of the conditions coming from each case, not their logical conjunction. Therefore you then have to consider, as solution set, the union of their solution sets (which correponds to the points that satisfy at least one of the two conditions).
Which, on a side note, is a reason why the condition $8x^2+x-9\ge0$ must also be added (this time by logical conjunction) to your conditions for case 2. Also, in case 1 it should be $-x+7x+2\ge 0$ (or whatever the LHS is), not $-x+7x+2>0$.
There might be some "bonus" mistakes of algebra in your work, as Barry was pointing out. I did not look for them, and I don't think the discussion is ripe for addressing them.
Assuming that the mistakes are yours and not from your book, I would say that the method you are being taught for transforming equations and inequations with square roots in systems of equations and inequations is generally fine for problems of this ilk, and personally I think it's best.