Dtyjlui

I have this question:

Just to check, this is the same as:

$$ -0.5 < sqrt{x+3} - left(frac{7}{4} + frac{x}{4} right) < 0.5 $$
$$ -.5 - sqrt{x+3} < - left(frac{7}{4} + frac{x}{4} right) < 0.5 - sqrt{x+3} $$

How do I go from here to get to the book's last line?

Negate your last line (and change the directions of both inequalities).

Alternatively, start with
$$-0.5 < frac{7}{4} + frac{x}{4} - sqrt{x+3} < 0.5$$
instead.

You can multiply all parts by $-1$
$$a>b$$
$$-a<-b$$

If $$|a-y|<b$$
then $y$ is from $a$ at most $b$, so $$a-b<y<a+b$$

In your case $a= sqrt{x+3}$, $b=0.5$ and $y=(x+7)/4$.

If you have $|M| < b$ then that means both

$-b < M < b$ and $-b < -M < b$ and you can use whichever one is useful[1].

So $|sqrt{x + 3} -(frac 74 +frac x4)| < 0.5$ can be manipulated in either of these two ways:

1) $-0.5 < sqrt{x+3} - (frac 74 + frac x4) < 0.5$

$-sqrt{x+3} - 0.5 < - (frac 74+frac x4) < - sqrt{x+3} +0.5$

... and now we can multiply everything by $-1$ and "flip the inequality" signs....

$(-1)(-sqrt{x+3} - 0.5) > (-1)(- (frac 74+frac x4)) > (-1)(- sqrt{x+3} +0.5)$

$sqrt{x+3} + 0.5 > frac 74 + frac x4 > sqrt{x+3}- .05$

$sqrt{x+3} - 0.5 < frac 74 + frac x4 <sqrt{x+3}+ .05$

or

2) $-0.5 < (frac 74 + frac x4) - sqrt{x+3} < 0.5$

$sqrt{x+3} -0.5 < frac 74 + frac x4 < sqrt{x+3} + 0.5$

Note... For $M = sqrt{x+3} - (frac 74 + frac x4)$ and $|M| < 0.5$ it was more useful to do 2) $0.5 < -M < 0.5$ and that left us with the positive $frac 74 + frac x4$ rather the negative $-(frac 74 + frac x4)$.

[1]

So why does $|M| < b$ mean BOTH $-b < M < b$?

Well, Case 1: $M ge 0$. Then $|M| = M$ and we know that $0 le M < b$. And so $-b < 0 le M < b$ so $-b < M < b$ and we also know that if $0 ge M < b$ then $-b < -M le 0 < b$ so $-b < -M < b$.

And Case 2: $M < 0$. Then $|M| = -M$ and $0 < -M < b$. And so $-b < 0 < -M < b$ but also $-b < M < 0$ so $-b <M < 0 < b$.

In bothe cases we will always have $-b < M < b$ AND we will also have $-b < -M < b$.

=====

Another, and probably better way, to think about it is greedoid's answer.

$|M -N|$ means "the positive distance between $M$ and $N$" so

$|M-N| < k$ means "$M$ and $N$ are $k$ distance within each other".

That means:

$N-k < M < N+k$ (in other words $M$ is within $k$ of $N$-- either within $k$ below or within $k$ above) and $M-k < N < M + k$ (in other words $N$ is within $k$ of $M$-- either within $k$ below or within $k$ above).

So $|sqrt{x+3}- (frac 74 + frac x4) < 0.5$ means that $sqrt{x+3}$ and $frac 74 + frac x4$ are within $0.5$ of each other.

So $frac 74 + frac x4$ is within $0.5$ of $sqrt{x+3}$. Either it is within $0.5$ below $sqrt{x+3}$ and $sqrt{x+3} - 0.5 < frac 74 + frac x4$. Or it is within $0.5$ above $sqrt{x+3}$ and $frac 74 + frac x4 < sqrt{x+3} + 0.5$.

Either way: $sqrt{x+3}- 0.5 < frac 74 + frac x4 < sqrt{x+3} + 0.5$.

$$-epsilon < dfrac{x+7}{4} - sqrt{x+3} < epsilon$$
$$-4epsilon < x+7 - 4sqrt{x+3} < 4epsilon$$
$$-4epsilon < (x+3) - 4sqrt{x+3} + 4 < 4epsilon$$
$$-4epsilon < (sqrt{x+3}-2)^2 < 4epsilon$$
$$(sqrt{x+3}-2)^2 < 4epsilon$$
$$-2sqrt epsilon < sqrt{x+3}-2 < 2sqrt epsilon$$
$$2-2sqrt epsilon < sqrt{x+3} < 2+2sqrt epsilon$$
$$(2-2sqrt epsilon)^2 < x+3 < (2+2sqrt epsilon)^2$$
$$(2-2sqrt epsilon)^2-3 < x < (2+2sqrt epsilon)^2-3$$

For $epsilon = frac 12$ we get $-2.66 < x < 8.66$

For $epsilon = frac{1}{10}$ we get $-1.13 < x < 3.93$