# Absolute value.

Definition 1.4.1   Let be a real number. The absolute value of is the number denoted by and defined as follows:
• If , then .
• If , then .

Example 1.4.2

• ;
• .
• If , then .
• If or , then .
• If , then .

Properties 1.4.3
1. .
2. .
3. .
4. Triangle Inequality:      .
5. .

Example 1.4.4   Write without absolute values.
 5 0 5 5 0 0 5 10 0 0 0
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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 .

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

 the equation solutions 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 coordinate of the unique point of intersection of these graphs.

Example 1.4.6   Solve the inequation .

We draw a table:

the inequation
 the solutions are given by

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Finally the solution set of the given inequation is .

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 coordinates for which the blue graph is higher than the red one.

Example 1.4.7   Solve the inequation .

We draw a table:

the inequation
 the solutions are given by

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i.e. the solution set of the proposed inequation is .

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 coordinates for which the blue graph is higher than the red one.

Noah Dana-Picard 2007-12-28