Absolute Value and Inequalities

Examples.

$$|-47| = 47, \quad\hbox{while}\quad |150| = 150 \quad\hbox{and}\quad |0| = 0.$$

$$-\left|-\dfrac{41}{7}\right| = -\dfrac{41}{7}.$$

The minus signs don't cancel; they're "blocked" by the absolute value.

Geometrically, the absolute value of a number is its distance from the origin 0. So -13.8 is $|-13.8| =
   13.8$ units from 0.

More generally, $|\hbox{foo} -
   \hbox{bar}|$ is the distance from foo to bar. For example,

$$\hbox{The distance from } -3 \hbox{ to } 5 \hbox{ is } |-3 - 5| = |-8| = 8.$$

$$\hbox{\epsfxsize=3in \epsffile{absval1.eps}}$$

$$\hbox{The distance from } 8 \hbox{ to } -9 \hbox{ is } |8 - (-9)| = |8 + 9| = |17| = 17.$$

$$\hbox{The distance from } -11 \hbox{ to } -39 \hbox{ is } |-11 - (-39)| = |-11 + 39| = |28| = 28.\quad\halmos$$


Example. Is the following equation true?

$$|6 - 4|\ \mathop{=}^{\rm ?}\ |6| - |4|$$

The equation is true:

$$|6 - 4| = |2| = 2, \quad\hbox{while}\quad |6| - |4| = 6 - 4 = 2.$$

So $|6 - 4| = |6| - |4|$ .


Example. Is the following equation true?

$$|-6 - 4|\ \mathop{=}^{\rm ?}\ |-6| - |4|$$

The equation is not true, because

$$|-6 - 4| = |-10| = 10, \quad\hbox{while}\quad |-6| - |4| = 6 - 4 = 2.$$

So $|-6 - 4| \ne |-6| - |4|$ .


Example. Is the following algebraic operation legal (for all a and b)?

$$|a - b|\ \mathop{=}^{\rm ?}\ |a| - |b|$$

In the last two examples, I saw one case where it works and another where it doesn't. Hence, the operation is not legal for all a and b (and in particular, it doesn't count as a rule of algebra).


Example. Is the following equation true?

$$|3\cdot (-5)|\ \mathop{=}^{\rm ?}\ |3|\cdot |-5|$$

The operation is valid:

$$|3\cdot (-5)| = |-15| = 15, \quad\hbox{while}\quad |3|\cdot |-5| = 3\cdot 5 = 15.$$

The two sides are equal.


Example. Simplify $\dfrac{|-2 - (-3)|}{6 - |-4|}$ .

In simplifying algebraic expressions, it's often helpful to work "from the inside out". First, I change $-(-3)$ to 3 and $|-4|$ to 4:

$$\dfrac{|-2 - (-3)|}{6 - |-4|} = \dfrac{|-2 + 3|}{6 - 4}.$$

Next, I do $-2 + 3 = 1$ and $6 - 4 =
   2$ :

$$\dfrac{|-2 + 3|}{6 - 4} = \dfrac{|1|}{2}.$$

Finally, $|1| = 1$ :

$$\dfrac{|1|}{2} = \dfrac{1}{2}.$$

That is,

$$\dfrac{|-2 - (-3)|}{6 - |-4|} = \dfrac{1}{2}.\quad\halmos$$


Example. Simplify $\dfrac{(-3)|4 - 7|}{|-3| - |4|}$ .

First, $4 - 7 = -3$ , $|-3| = 3$ , and $|4| =
   4$ :

$$\dfrac{(-3)|4 - 7|}{|-3| - |4|} = \dfrac{(-3)|-3|}{3 - 4}.$$

(Notice again how I'm evaluating the expression "inside out".)

Next, $|-3| = 3$ and $3 - 4 = -1$ :

$$\dfrac{(-3)|-3|}{3 - 4} = \dfrac{(-3)\cdot 3}{-1}.$$

The rest is just arithmetic:

$$\dfrac{(-3)\cdot 3}{-1} = \dfrac{-9}{-1} = 9.\quad\halmos$$


Example. Graph the inequality $1 < x < 5$ .

$$\hbox{\epsfxsize=3in \epsffile{absval2.eps}}\quad\halmos$$


Example. Graph the inequality $x \ge -7$ .

$$\hbox{\epsfxsize=3in \epsffile{absval3.eps}}\quad\halmos$$


Example. What inequality is represented by the following picture?

$$\hbox{\epsfxsize=3in \epsffile{absval4.eps}}$$

$-4 < x \le 2$ . The open (white) circle means the point -4 is {\it not} included; the closed (black) circle means the point 2 is included.


Example. What inequality is represented by the following picture?

$$\hbox{\epsfxsize=3in \epsffile{absval5.eps}}$$

Here is an easy way to write down inequalities from number line pictures. First, put $<$ 's on either side of each number. Be sure they're $<$ 's, not $>$ 's (i.e. be sure they all point the same way).

$$\hbox{\epsfxsize=3in \epsffile{absval6.eps}}$$

Now put an x under each shaded part, and pick off the inequalities corresponding to the shaded parts: $x <
   -7$ or $-1 < x$ .

But note that you cannot write this as "$-1 < x < -7$ ", because (ignoring the x), this says "$-1 < -7$ ", which is {\it false}.


Example. Which is bigger, $\dfrac{2}{3}$ or $\dfrac{4}{7}$ ?

There are several reasonable ways to figure this out. One approach is to compute the decimal values of these fractions. Since $\dfrac{2}{3} \approx 0.66667$ while $\dfrac{4}{7} \approx 0.57143$ , it follows that $\dfrac{2}{3} > \dfrac{4}{7}$ .

You can compute the decimal values with a calculator, but also by hand, using long division.


Example. Which is bigger, $-\dfrac{3}{5}$ or $-\dfrac{2}{3}$ ?

Since $-\dfrac{3}{5} = -0.6$ while $-\dfrac{2}{3} \approx -0.66667$ , it follows that $-\dfrac{3}{5} > -\dfrac{2}{3}$ .



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