How do you solve #abs(8x+8) = abs(9x-4)#?

Question

Abigail Allen · Accepted Answer

#abs(u) = abs(v)# if and only if either #u=v# or #u=-v#.

Two numbers have the same absolute value when the numbers are equal or they are opposites (negatives) of each other. #abs(8x+8) = abs(9x-4)# if and only if: #8x+8 = 9x-4# #color(white)"xx"# or #color(white)"xx"# #8x+8 = -(9x-4)# Solving #8x+8 = 9x-4# , we get #8+4 = 9x-9x# so #12=x#, that is: #x=12# Solving #8x+8 = -(9x-4)#. we get #8x+8 = -(9x-4)#, so #8x+8 = -9x+4# and then #8x+9x = 4-8#, so #17x = -4#, and finally #x = -4/17# The solutions are: #12# and #-4/17#

Abigail Allen · Answer

Split into cases for intervals #(-oo, -1)#, #[-1, 4/9)# and #(4/9, oo)#, solving the linear equations that result to find:

#x=-4/17# or #x=12# Split into cases: Case 1: #x in (-oo, -1)# #8x+8 < 0#, so #abs(8x+8) = -(8x+8) = -8x-8# #9x-4 < 0#, so #abs(9x-4) = -(9x-4) = -9x+4# Equation becomes: #-8x-8 = -9x+4# Add #9x+8# to both sides to get: #x = 12# This lies outside #(-oo, -1)# so is not valid for this case. Case 2: #x in [-1, 4/9)# #8x+8 >= 0#, so #abs(8x+8) = 8x+8# #9x-4 < 0#, so #abs(9x-4) = -(9x-4) = -9x+4# Equation becomes: #8x+8 = -9x+4# Add #9x-8# to both sides to get: #17x=-4# Divide both sides by #17# to get: #x=-4/17# This does lie in #[-1, 4/9)# so is a valid solution. Case 3: #x in [4/9, oo)# #8x+8 >= 0#, so #abs(8x+8) = 8x+8# #9x-4 >= 0#, so #abs(9x-4) = 9x-4# Equation becomes: #8x+8 = 9x-4# Add #-8x + 4# to both sides to get: #x = 12# This does lie in #[4/9, oo)# so is a valid solution.

Jace Anderson · Answer

undefined To solve the equation $ |8x + 8| = |9x - 4| $, you need to consider two cases:

1. When $ 8x + 8 $ and $ 9x - 4 $ are both positive or both negative.
2. When $ 8x + 8 $ is positive and $ 9x - 4 $ is negative, or vice versa.

For the first case:

$ 8x + 8 = 9x - 4 $
Solve for $ x $.

For the second case:

$ 8x + 8 = -(9x - 4) $
Solve for $ x $.

Then, check the solutions to ensure they satisfy the original equation.