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o Step 3: set the predominant sign for each
term under each condition, and solve the
equation. In the example.

Absolute value
• Formulas to remember:

o By definition of |x|, |x| = x if x ≥ 0 and |x|
= -x if x < 0
• Absolute value exercises like: How many
solutions will this equation have? |x+3| – |4-x|
= |8+x| ---> we apply the "Critical Values"
method for absolute values equations:
o Step 1: do each absolute term equal to zero
to obtain critical values. In the example,
critical values are -3, 4, -8.
o Step 2: place them in the Real numbers line
to make all the possible intervals. Note that
you have to include critical values in the
intervals, that is why we put the term "less
or equal" and "big or equal"... we put them
as below by convention. In the example, the
intervals (or conditions) are:

Condition 1: -(x+3) - (4-x) = -(8+x) --> x = -1
Condition 2: -(x+3) - (4-x) = (8+x) --> x = -15
Condition 3: (x+3) - (4-x) = (8+x) --> x = 9
Condition 4: (x+3) + (4-x) = (8+x) --> x = -1

o Step 4: check if the solution satisfies the
initial condition. In the example:

Condition 1: NO, -1 is not less than -8
Condition 2: NO, -15 is not between -8 and -3
(including -8 in the interval and not -3)
Condition 3: NO, 9 is not between -3 and 4
(including -3 in the interval and not 4)
Condition 4: NO, -1 is...

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