Home > Algebra > Absolute Value Equations

How do you solve #|x + 6| = 2x# and find any extraneous solutions?

Answer 1

Kennedy Adams

#x=-2# if #x<-6#
#x=6# if#x>=-6#

Knowing the property of absolute value that says:

if#|x|=a# then #x=-a , x<0#

#x=a,x>=0#

Applying the above property we have:

#x+6=-2x,x+6<0#. Eq(1)

#x+6=2x,x+6>=0#. Eq(2)

Solving the above equations we have:

Eq(1): if #x+6<0# that is #x<-6# #x+2x=-6# #3x=-6# #x=-6/3=-2# So, #x=-2,x<-6#

Eq(2): if #x+6>=0# that is #x>=-6#

#x+6=2x# #x-2x=-6# #-x=-6# #x=6# whenever #x>=-6#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Jace Anderson

To solve the equation |x + 6| = 2x and identify any extraneous solutions, follow these steps:

Solve for x when x + 6 is positive: x + 6 = 2x x = 6
Solve for x when x + 6 is negative: -(x + 6) = 2x -x - 6 = 2x -3x = 6 x = -2

Check for extraneous solutions by substituting these values back into the original equation:

For x = 6: |6 + 6| = 2(6) |12| = 12 (True)

For x = -2: |-2 + 6| = 2(-2) |4| = -4 (False, since absolute value cannot be negative)

Therefore, the extraneous solution is x = -2. The valid solution is x = 6.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.