[Math] Absolute value equations? (I keep getting the wrong answer)

solve algebraically: x = |4x-1|

txtbook answer: 1/3 and 1/5
my answer: No solution (i got 1/4 for both cases but they are both extraneous roots)

Thank you so much!

When working with absolute values, remember that the quantity inside the | | has two values, one positive and one negative, so there will be two possible solutions. Set the quantity to each value and solve for the unknown.

x = |4x - 1|
x = ± (4x - 1)

First Possible Solution:

x = 4x - 1
x - 4x = - 1
- 3x = - 1
x = - 1 / - 3
x = 1/3
¯¯¯¯¯¯

Second Possible Solution:

x = - (4x - 1)
x = - 4x + 1
x + 4x = 1
5x = 1
x = 1/5
¯¯¯¯¯¯
 

The absolute value symbol is, in effect, a multiplier by +1 or -1. So an equation such as this is solved as two separate equations.
[equation 1] .. x = +1(4x-1)
.. -3x = -1 ... subtract 4x
.. x = 1/3 ... divide by -3
Check: 1/3 = |4(1/3) - 1| = |4/3 - 1| = |1/3| ... yes

[equation 2] .. x = -1(4x-1)
.. 5x = 1 ... add 4x
.. x = 1/5 ... divide by 5
Check: 1/5 = |4(1/5) - 1| = |4/5 - 1| = |-1/5| ... yes

x=|4x-1| implies x^2=(4x-1)^2 and therefore
15x^2-8x+1=(5x-1)*(3x-1)=0.
Solutions of this equation are 1/5 and 1/3. Plugging in the original equation we see that both work (squaring could add additional solutions!).

| 4x–1 | = x

Consider | ? | = x if and only if, ? =x or ? = –x right? So

4x – 1 = x OR 4x – 1 = – x Solve each of these: (read each vertically)

4x – x = 1 OR 4x + x = 1

3x = 1 OR 5x = 1

x = 1/3 OR x = 1/5

Honestly i have no clue, but i bet the guys at confusinghomework.com could help you out. They show work and the best part is its FREE!!!