Absolute value.

Definition 1.4.1   Let $ x$ be a real number. The absolute value of $ x$ is the number denoted by $ \vert x\vert$ and defined as follows:

Example 1.4.2       

Properties 1.4.3  
  1. $ \forall x \in \mathbb{R}, \vert x\vert=0 \Longleftrightarrow x=0$ .
  2. $ \forall x \in \mathbb{R}, \; \vert-x\vert=\vert x\vert$ .
  3. $ \forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \vert xy\vert=\vert x\vert \cdot \vert y\vert$ .
  4. Triangle Inequality:      $ \forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \vert x+y\vert \leq \vert x\vert + \vert y\vert$ .
  5. $ \forall x \in \mathbb{R}, \forall y \in \mathbb{R}^+, -y \leq x \leq y \Longleftrightarrow \vert x\vert \leq y$ .

Example 1.4.4   Write $ f(x)=\vert 2x-1\vert+\vert x-3\vert-\vert x+2\vert$ without absolute values.
$ x$ $ - \infty \qquad \qquad$ $ -2$ $ \qquad \qquad $ $ 1/2$ $ \qquad \qquad $ $ 3$ $ \qquad \qquad + \infty$
$ \vert 2x-1\vert$ $ 1-2x$ 5 $ 1-2x$ 0 $ 2x-1$ 5 $ 2x-1$
$ \vert x-3\vert$ $ 3-x$ 5 $ 3-x$ $ 5/2$ $ 3-x$ 0 $ x-3$
$ \vert x+2\vert$ $ -x-2$ 0 $ x+2$ $ 5/2$ $ x+2$ 5 $ x+2$
$ f(x)$ $ 7-2x$ 10 $ 2-4x$ 0 0 0 $ 2x-7$
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In the following examples, we develop the algebraic solution process; the reader can have some help from the drawing of the appropriate graphs. The reader who is not familiar graphs can skip to chapter 2 subsection graph of a function and return to the present section afterwards.

Example 1.4.5   Solve the equation $ \vert x-2\vert=2x+3 -\vert x-1\vert$ .

Figure 2:
\begin{figure}\mbox{\epsfig{file=AbsoluteValue-Equation.eps,height=5cm}}\end{figure}

Let us draw a table, in order to write the different expressions without an absolute value:

$ x$ $ - \infty \qquad \qquad$ $ 1$ $ \qquad \qquad $ $ 2$ $ \qquad \qquad + \infty$    
$ \vert x-2\vert$ $ -x+2$      $ -x+2$      $ x-2$    
$ \vert x-1\vert$ $ -x+1$      $ x-1$      $ x-1$    
the equation $ -x+2=3x+2$      $ -x+2=x+4$      $ x-2=x+2$    
solutions $ x=0$      none      none    
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The proposed equation has 0 as its unique solution.

Look at Figure 2: the equation's left side corresponds to the blue graph, the right side corresponds to the red graph. The solution of the equation is the $ x-$ coordinate of the unique point of intersection of these graphs.

Example 1.4.6   Solve the inequation $ \vert 2x-3\vert \geq \vert x+1\vert$ .

We draw a table:

$ x$ $ - \infty \qquad \qquad$ $ -1$ $ \qquad \qquad $ $ 3/2$ $ \qquad \qquad + \infty$    
$ \vert 2x-3\vert$ $ -2x+3$      $ -2x+3$      $ 2x-3$    
$ \vert x+1\vert$ $ -x-1$      $ x+1$      $ x+1$    
the inequation $ -x \geq -4$      $ -3x \geq -2 $      $ x \geq 4$    
the solutions are given by
$ x \leq -1$      $ -1 \leq x \leq 3/2$      $ x \geq 4$    
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Finally the solution set of the given inequation is $ (-\infty, 2/3 ] \cup [4,+\infty)$ .

Figure 3:
\begin{figure}\mbox{\epsfig{file=AbsoluteValue-Inequation1.eps,height=5cm}}\end{figure}

Look at Figure 3: the inequation's left side corresponds to the blue graph, the right side corresponds to the red graph. The solution set of the inequation is the set of $ x-$ coordinates for which the blue graph is higher than the red one.

Example 1.4.7   Solve the inequation $ \vert 2x-3\vert \geq x+\vert x+1\vert$ .

We draw a table:

$ x$ $ - \infty \qquad \qquad$ $ -1$ $ \qquad \qquad $ $ 3/2$ $ \qquad \qquad + \infty$    
$ \vert 2x-3\vert$ $ -2x+3$      $ -2x+3$      $ 2x-3$    
$ \vert x+1\vert$ $ -x-1$      $ x+1$      $ x+1$    
the inequation $ -2x-3 \geq -1$      $ -2x+3 \geq 2x+1$      $ 2x-3 \geq 2x+1$    
the solutions are given by
$ x \leq 2$      $ x \leq 1/2$      $ -3 \geq 1$    
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i.e. the solution set of the proposed inequation is $ (-\infty, 1/2]$ .

Now look at Figure 4: the inequation's left side corresponds to the blue graph, the right side corresponds to the red graph. The solution set is the set of $ x-$ coordinates for which the blue graph is higher than the red one.

Figure 4:
\begin{figure}\mbox{\epsfig{file=AbsoluteValue-Inequation2.eps,height=5cm}}\end{figure}

Noah Dana-Picard 2007-12-28