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EXAMPLES & SOLUTIONS
Anchor 1
Two algebraic expressions that contain at least one absolute value to solve a problem.
The symbol sign (=) states that it is an equation.
Absolute values always equal to a positive number or zero.
Absolute value equations may have two solutions, one solutions or no solution.
Absolute equations with two solutions are compared to a postitive number.
Absolute value equations that are equal to zero have one solution.
One-variable absolute value equations can have more than one answer, but only one solution.
Absolute value equations that are less than or equal to a negative number have no solution.
ABSOLUTE VALUE EQUATION
Explanation of absolute value equations
expression: | x - 5 | = 3
Because we cannot know if | x | is neg. or pos, we must compare the number by making it neg. or pos. to discover the possible solutions.
Solve:
| x - 5 | = -3 --absolute value?
| x - 5 | = x - 5
x - 5 = -3 --updated equation
x - 5 + 5 = -3 + 5 --add 5 from both sides
x + 0 = 2
x = 2
| x - 5 | = 3 --absolute value?
| x - 5 | = x - 5
x - 5 = 3 --updated equation
x - 5 + 5 = 3 + 5 --add 5 from both sides
x + 0 = 8
x = 8
Solve:
Fact: It is always good to verify (check) your answer.
Follow properties and PEMDAS:
Check:
| x - 5 | = ?
| x - 5 | = 3
| x - 5 | = 3 --substitute 2 for x
| 2 - 5 | = 3
| -3 | = 3 --absolute value?
3 = 3 --CORRECT
x = 2 --Solution
| x - 5 | = ?
| x - 5 | = 3
| x - 5 | = 3 --substitute 8 for x
| 8 - 5 | = 3
| 3 | = 3 --absolute value?
3 = 3 --CORRECT
x = 8 --Solution
Follow properties and PEMDAS:
Check:
Absolute Value Equations
Examples of absolute value equations with one solution
expression: | x - 5 | = 0
We must compare the number zero by making it neg. or pos.
Since -0 and +0 are the same ( + 0 = 0), then there is only one solution.
Solve:
Follow properties and PEMDAS:
| x - 5 | = 0 --absolute value?
| x - 5 | = x - 5
x - 5 = 0 --updated equation
x - 5 + 5 = 0 + 5 --subtract 5 from both sides
x + 0 = 5
x = 5
Solve:
Follow properties and PEMDAS:
| x - 5 | = 0 --absolute value?
| x - 5 | = x - 5
x - 5 = -0 --updated equation (-0 = 0)
x - 5 + 5 = 0 + 5 --add 5 from both sides
x + 0 = 5
x = 5
Check:
| x - 5 | = 0 --absolute value?
| x - 5 | = x - 5
x - 5 = 0 --updated equation
5 - 5 = 0 --subtract 5 from both sides
0 = 0 --CORRECT
x = 5 --Solution
Fact: Sometimes you can get an answer to a absolute value equation, but have no solution.
Absolute Value Equations
Examples of absolute value equations no solution
expression: 5 = | x + 2 | + 8
Solve:
5 = | x + 2 | + 8
5 - 8 = | x + 2 | + 8 - 8 --substract -8 from both sides
-3 = | x + 2 | + 0
-3 = | x + 2 |
| x + 2 | = x + 2 --absolute value?
-3 = x + 2 --updated expression
-3 - 2 = x + 2 - 2 --subtract -2 from both sides
-3 - 2 = x + 0 --subtract rule
-5 = x
Check:
5 = | x + 2 | + 8
5 = | -5 + 2 | + 8
| -5 + 2 | = | -3 | = 3 --absolute value?
5 = 3 + 8
5 = 11 --INCORRECT
x = -5 --Not a solution
Solve:
5 = | x + 2 | + 8
5 - 8 = | x + 2 | + 8 - 8 --substract -8 from both sides
-3 = | x + 2 | + 0
-3 = | x + 2 |
| x + 2 | = x + 2 --absolute value?
-(-3) = x + 2 --updated expression
3 = x + 2
3 - 2 = x + 2 - 2 --subtract -2 from both sides
3 - 2 = x + 2 - 2 --subtract -2 from both sides
1 = x
Check:
5 = | x + 2 | + 8
5 = | 1 + 2 | + 8
| 1 + 2 | = | 3 | = 3 --absolute value?
5 = 3 + 8
5 = 11 --INCORRECT
x = 1 --Not a solution
Always simplify the equation before solving for the absoute value.
Absolute Value Equations
Example with absolute value equations with two solution
expression: | 2x - 4 | + 8 = 10
Solve:
| 2x -4 | + 8 = 10 --add 2 to both sides
| 2x -4 | + 8 - 8 = 10 - 8 --combine like terms
| 2x - 4 | - 0 = 2
| 2x - 4 | = 2x - 4 --absolute value?
2x - 4 = (+2) --updated equation
2x - 4 + 4 = 2 + 4 --subtract 4 from both sides
2x + 0 = 6
2x = 6
2x = 6 --updated expression
-2 -2
x = 3
Check:
| 2x -4 | + 8 = 10 --original equation
| 2 (3) -4 | + 8 = 10 --plug in solution, 3, and multiply
| 6 - 4 | + 8 = 10
| 6 - 4 | = | 2 | = 2 --absolute value?
2 + 8 = 10
10 = 10
x = 3 --Solution
Solve:
| 2x -4 | + 8 = 10 --add 2 to both sides
| 2x -4 | + 8 - 8 = 10 - 8 --combine like terms
| 2x - 4 | - 0 = 2
| 2x - 4 | = 2x - 4 --absolute value?
2x - 4 = (-2) --updated equation
2x - 4 + 4 = -2 + 4 --subtract 4 from both sides
2x + 0 = 2
2x = 2
2x = 2 --updated expression
2 2
x = 1
Check:
| 2x -4 | + 8 = 10 --original equation
| 2 (1) -4 | + 8 = 10 --plug in solution, 1, and multiply
| 2 - 4 | + 8 = 10
| 2 - 4 | = | -2 | =2 --absolute value?
2 + 8 = 10
10 = 10
x = 1 --Solution
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