So I'm trying to solve: $|4x+3| > |x|$
I've broken each absolute function down and tried solving it for each condition of $x$, as indicated in the picture. The answer is $x < -1$ or $x > -3/5$. The first and third statements corroborate with this conclusion, but the second seems to contradict the first. Am I doing something wrong?
The second statement is that $x > -1$ WHEN $x \ge 0$. But since $x \ge 0$ anyway, the conclusion is just that $x \ge 0$.
Note, we actually need this. As the conclusion $x > -\frac 35$ assumed $x < 0$. We need to consider what happens with $x \ge 0$. And what happens is... $x \ge 0$....
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Your statments aren't just the conclusions. They are the conclusions combined with the hypotheses.
So 1) is: If $x < -\frac 34$ then $x < -1$. So both $x < -\frac 34$ and $x < -1$ would be true. Combining them we get $x < -1 < -\frac 34$ or simply $x< -1$. ($x < -\frac 34$ is redundant.)
2) is: If $x \ge 0$ then $x > -1$. So both $x \ge 0$ and $x > -1$ would be true. Combining them we get $x \ge 0 > -1$ or simply $x \ge 0$. (Our conclusion was a bust! We didn't learn anything! Oh well, it happens.)
3) is: If $-\frac 34 \le x < 0$ then $x> -\frac 35$. So combining them we get $-\frac 34\le -\frac 35 < x < 0$.
So we get
$x< -1$ or $-\frac 35 < x < 0$ or $x \ge 0$.
We can combine the last two to get
$x < -1$ or $x > -\frac 35$.
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The contradict each other because you are making different assumptions. That shouldn't be surprising. It it means one or the other is true. And as it turns out, they can't be both true. But that's fine because your HYPHOTHESES can't both be true. Either $x < -1$ and one thing happens. Or $x \ge -1$ and ... something else happens.
That isn't a problem.
Suppose I figured either Batman was Bruce Wayne or Batman was Clark Kent. If Batman was Bruce Wayne, I conclude that $2$ is a prime number, Gotham City is on the East Coast and Gamputters are soft and fuzzy. If Batman is Clark Kent, I conclude that $2$ is a prime number, Gotham City is in the Midwest, and Gamputters are sharp and prickly.
These are contradictory. Where did I go wrong? Nowhere, they are contradictory because Batman being both Bruce Wayne AND Clark Kent is contradictory.
So in this case one of them is true OR one of the others is true depending upon which of the contradictory cases is is true.
IF $x < -\frac 34$ then $x < -1$.
IF $x \ge 0$ (which mean $x < -\frac 34$ is FALSE) then $x >-1$. This contradicts $x < -1$ but that was only true if $x<-\frac 34$ which is NOT THE CASE if $x \ge 0$.
IF $-\frac 34 \le x < 0$ then $x > -\frac 35$. This also contracicts $x < -1$ but since we are assuming $x$ is NOT less than $-\frac 34$ this doesn't matter.
So there are three possiblities.
Either
1) $x < \frac 34$ and $x < -1$ so $x < -1$
OR
2) $x \ge 0$ and $x > -1$ so $x \ge 0$.
OR
3) $-\frac 34 \le x < 0$ and $x > -\frac 35$ so $-\frac 35 < x < 0$
So
$x < -1$ or $x \ge 0$ or $-\frac 35 < x < 0$.
So $|4x + 3| > |x|$ is true if
$$x < -1\text { or }x > -\frac 35$$.
And $|4x+3| > |x|$ is false if
$$ -1\le x \le -\frac 35$$
[This is all assuming that your math was correct. I did not check it.]