# Absolute value inequality, where am I wrong?

Tried to solving $|x^2-5x+5|<1$ using the square method, but I don't know what I did wrong:

$$-1<x^2-5x+5<1$$ $$-6<x^2-5x<-4$$ $$-6+\frac{25}{4}<x^2-5x+\frac{25}{4}<-4+\frac{25}{4}$$ $$\frac{25-24}{4}<\left(x-\frac{5}{2}\right)^2<\frac{25-16}{4}$$ $$\frac{\pm\sqrt1}{\sqrt4}<\sqrt{\left(x-\frac{5}{2}\right)^2}<\frac{\pm\sqrt9}{\sqrt4}$$ $$\frac{\pm1}{2}<x-\frac{5}{2}<\frac{\pm3}{2}$$ $$\frac{5\pm1}{2}<x<\frac{5\pm3}{2}$$

Possible solutions:

$2<x<1$ (not valid)

$2<x<4$ (ok)

$3<x<1$ (not valid)

$3<x<4$ (ok, but is a subset of solution 2.

Therefore $S=\{2<x<4\}$

The only problem is that the correct solution is $S=\{1<x<2\text{ or }3<x<4\}$. Where am I wrong?

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Hint: what happens when you take square roots in an inequality and consider both the positive and negative root. For example, $1<4<9$, but taking square roots, $\pm 1<2<\pm 3$ gives something fishy. Afterall, if say $-1<0$ but squaring both sides does not preserve the inequality: $1<0$. – Alex R. May 3 '12 at 20:25
The double inequality $$\frac{25-24}{4}<\left(x-\frac{5}{2}\right)^2<\frac{25-16}{4}$$ is not equivalent to $$\frac{\pm\sqrt1}{\sqrt4}<\sqrt{\left(x-\frac{5}{2}\right)^2}<\frac{\pm\sqrt9}{‌​\sqrt4}$$ – Américo Tavares May 3 '12 at 20:29
Why not? If I had them separated (like equation 1 and 2) wouldn't it be valid? I'm not questioning your math, I'm just trying to grasp it. – Luiz Borges May 3 '12 at 20:32
@Luiz Borges Well, $\sqrt{\left( x-\frac{5}{2}\right) ^{2}}\geq 0$ and $\frac{-\sqrt{9}}{\sqrt{4}}<0$; and similarly for the first inequality. – Américo Tavares May 3 '12 at 21:07
@LuizBorges The solution is $1<x<2$ or $3<x<4$. Proof. The solution of the first inequality $x^{2}-5x+4<0$ is $1<x<4$, because the roots of $x^{2}-5x+4=0$ are $x_{1}=1$ and $x_{2}=4$, and the coefficient of $x^{2}$ is positive. The solution of the second inequality $x^{2}-5x+6>0$ is $x<2$ or $x>3$, because the roots of $x^{2}-5x+6=0$ are $x_{1}=2$ and $x_{2}=3$, and the coefficient of $x^{2}$ is positive. – Américo Tavares May 3 '12 at 21:15

You have a $\pm$ on each side of the inequality, but you need to change the direction of inequality for the "minus".

So you would have $$\dfrac{5 + 1}{2} < x < \dfrac{5+3}{2}$$ (The $+$'s go together), or $$\dfrac{5 - 1}{2} > x > \dfrac{5 - 3}{2}$$ (The $-$'s go together)

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I did reached that by trial analysis, but why is that? I'm trying to run away from "magic rules"... :( – Luiz Borges May 3 '12 at 20:31
See Phira's answer, which I will now comment on. – The Chaz 2.0 May 3 '12 at 20:32
I got it know. I will accept your answer, but it could be Phira as well. I got choose one. Thanks. – Luiz Borges May 3 '12 at 20:54
Anytime. Maybe I'll have a look at her other answers and find a few I haven't voted on :) – The Chaz 2.0 May 3 '12 at 21:00
@LuizBorges: It's not a "magic rule", it's the fact that $\sqrt{x^2}=|x|$, not $\pm x$. – Arturo Magidin May 3 '12 at 21:31

When you take the root of an inequality, you have to make sure that everything is positive and then take positive roots.

So after taking the roots, you get:

$\frac 12 < |x-\frac 52| < \frac 32$.

Now, you can regard the two cases $x> \frac 52$ and $x < \frac 52$ to eliminate the absolute value.

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(For Luiz) - When $x > \dfrac{5}{2}$, the absolute value can be dropped. This gives the first inequality in my answer. When $x < \dfrac{5}{2}$, you must multiply everything by $-1$, giving the second inequality in my answer. – The Chaz 2.0 May 3 '12 at 20:33