Absolute value equations
What number is x when | x | = 4?
According to the definition of absolute value, x is a number whose distance from zero is 4. So x = 4 or x = -4. An absolute value equation usually gives two solution paths.
| 2x | = 4
2x = 4 or 2x = -4, which gives x = 2 or x = -2.
| 3x-6 | = 9
3x-6 = 9 or 3x-6 = -9 to give x = 5 or x = -1
| 4x-7 | = -5
The absolute value is always greater than or equal to 0, so this equation has no solutions.
By the definition we get 2 equations
Two quadratic equations are obtained. The solutions of the former are x = -2 or x = 3 and the solutions of the latter are x = -1 or x = 2. So the solutions of the original absolute value equation are x = -2, x = -1, x = 2 or x = 3
An equation in which the absolute values of two functions are equal is true when the values of the functions are equal or opposite.
Solve the equation
|2x - 4| = |x - 5|
By definition we get two equations
2x - 4 = x - 5 or 2x - 4 = - (x - 5)
The first equation gives x = -1 and the second x = 3
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