1. Solving Absolute Value Equations Solving Absolute Value Equations
Absolute Values
Solving Absolute Value Equations
Not all functions are defined in a simple way by a neat formula. Sometimes we have to break up the domain and define the function differently on different pieces of the domain.
Absolute Value Equations 1
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2. Absolute Value Equations
Solving Absolute Value Equations
General Procedure
Form is:
for constants a, b and k ≥ 0
By definition:
| ax + b | = k
either | ax + b | = ax + b
= k OR
| ax + b | = –(ax + b)
= k
Solving Absolute Value Equations
The simplest way to solve equations with absolute values is to replace the absolute value terms with their equivalent. We appeal to the definition of absolute value.
For any number a we define | a | as
a , for a ≥ | a |
–a , for a < 0
Similarly if | a x + b | = k for some k ≥ 0, then a x + b = ±k.
Note: The solutions should be substituted into the original equation to verify that they produce a true statement.
(ax + b) = –k
ax + b = k + –
Thus =
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Absolute Value Equations
Absolute Value Equations
10/2/2013

3. Absolute Value Equations
Solving Absolute Value Equations
Example 1
Solve –3x – 2= 5
–3x – 2 = or –3x – 2 = –5
–3x = or –3x = –5 + 2
x = –7/3 or x = 1
Solution set is: { –7/3, 1 }
Solving Absolute Value Equations: Example 1
We replace the absolute value terms with their equivalent by appealing to the definition of absolute value.
For any number a we define | a | as
a , for a ≥ | a |
–a , for a < 0
Similarly if | a x + b | = k for some k ≥ 0, then a x + b = ±k.
In Example 1, k = 5, so either -3x – 2 = 5 or -3x – 2 = -5. We solve both inequalities revealing that either x = -7/3 or x = 1 as the two solutions for the original equation.
The solution set is thus { -7/3, 1 }.
Note: The solutions should be substituted into the original equation to verify that they produce a true statement. =
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Absolute Value Equations
Absolute Value Equations
10/2/2013

4. Absolute Value Equations
Solving Absolute Value Equations
Example 2
Solve 3x= –6
Note: 3x ≥ 0 for all x
Hence, 3x= –6
is FALSE for every value of x
Thus there is no solution and thus
solution set is
Solving Absolute Value Equations: Example 2
We apply the principles we have just discussed – along with common sense – in solving some absolute-value equations.
In Example 2 we simply apply common sense by noting that | 3x | is always greater than or equal to 0. So | 3x | can never equal -6, or any other negative number ! Hence this equation has no solution and the solution set is empty, that is { } or ∅. O
or { }
Note: O ≠ { }
WHY ?
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Absolute Value Equations
Absolute Value Equations
10/2/2013

5. Absolute Value Equations
Solving Absolute Value Equations
Example 3
Solve 2x – 3 – 4 = 7
2x – 3 = = 11
2x – 3 = 11 or –(2x – 3 ) = 11
2x – 3 = 11 or x – 3 = –11
2x = 14 or x = –8
x = or x = –4
Solution set is { –4, 7 }
Solving Absolute Value Equations: Example 3
We apply the principles we have just discussed – along with common sense – in solving some absolute-value equations.
In Example 3, in order to apply the rule that | a x + b | = ±k for some constant k, we must transform the equation into this form by adding 4 to both sides to get
| 2x – 3 | = 11
which in turn gives two equations:
2x – 3 = and 2x – 3 = -11
Solving these gives solutions 7 and -4, respectively. So solution set is { -4, 7}.
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Absolute Value Equations
Absolute Value Equations
10/2/2013

6. Absolute Value Equations
Solving Absolute Value Equations
Example 4
Solve 9 + x= 3 – 2x
Removing one absolute value at a time:
9 + x = 3 – 2x and 9 + x= –(3 – 2x)
Same
9 + x = 3 – 2x
–(9 + x) = –(3 – 2x)
Solving Absolute Value Equations: Example 4
Here we have an equation with two absolute values, one on each side of the equation. The strategy is to remove the absolute values one at a time. Recall that | a | is either a or –a , depending on whether a < 0 or a ≥ 0.
We remove the absolute values on the right side first. This gives either 3 – 2x or –(3 – 2x) on the right, both equal to the absolute value on the left.
Then we apply the definition of absolute value to remove the absolute value on the left from these two equations. In the first of the two we get either
9 + x = 3 – 2x or –(9 + x) = 3 – 2x
and in the second equation we get either
–(9 + x) = –(3 – 2x) or x = –(3 – 2x)
and we note that of these four equations the left hand two are equivalent and the right hand two are equivalent, as shown in the illustration.
In effect the equation with two absolute values is solved exactly like an equation with only one side in absolute value.
Here again we emphasize the importance understanding and using the definition of absolute value.
–(9 + x) = 3 – 2x
9 + x = –(3 – 2x)
Same
9 + x = –(3 – 2x)
9 + x = 3 – 2x
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Absolute Value Equations
Absolute Value Equations
10/2/2013

7. Absolute Value Equations
Solving Absolute Value Equations
Solve 9 + x= 3 – 2x
9 + x = –(3 – 2x)
9 + x = 3 – 2x
9 + x = 3 – 2x
or x = –3 + 2x
2x + x = 3 – 9
or = –x + 2x
3x = –6
or = x
Solving Absolute Value Equations: Example 4 (continued)
Here we have eliminated duplicate equations and narrowed the search to the two equations
9 + x = 3 – 2x and x = –(3 – 2x)
These two equations are equivalent to the original absolute value equation, so their solutions are the solutions of the original equation.
Solving these equations yields two solutions, namely -2 and 12 and so the solution set for the original equation is {-2, 12}.
In effect the equation with two absolute values is solved exactly like an equation with only one side in absolute value.
Here again we emphasize the importance understanding and using the definition of absolute value.
x = –2
or x = 12
Solution set is { –2, 12 }
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Absolute Value Equations
Absolute Value Equations
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8. Solving Absolute Value Equations
Think about it !
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Absolute Value Equations
Absolute Value Equations
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