How do you solve #abs(5.2x+7)=3.8#?

Question

Savannah Anderson · Accepted Answer

#x = -8/13, -27/13# CASE 1. #color(red)(5.2x+7 >= 0)# if #5.2x+7 >= 0# #i.e. x >= -7/5.2# or #x >= -35/26# then #|5.2x+7| = 5.2x+7# because the modulus of a positive number is the number itself. #therefore # in this case #5.2x+7=3.8# #=> 5.2x = -3.2# #=> x=-3.2/5.2 = color(red)(-8/13)# CASE 2. #color(red)(5.2x+7 < 0)# if #5.2x+7 < 0# #i.e. x < -7/5.2# or #x < -35/26# then #|5.2x+7| = -(5.2x+7) = -5.2x-7# because the modulus of a negative number is negative of the number or in other words modulus of a negative number is obtained by multiplying the number with #-1#. #e.g. -2<0 => |-2| = -(-2) or -1*(-2) = 2# #therefore # in this case #-5.2x-7=3.8# #=>-5.2x = 10.8# #=> x=-10.8/5.2 = color(red)(-27/13)# Hence, #x = color(red)(-8/13) or color(red)(-27/13)#.

Nathan Axel · Answer

undefined To solve the equation abs(5.2x + 7) = 3.8, you set up two equations by considering the positive and negative cases of the absolute value:

1. 5.2x + 7 = 3.8
2. -(5.2x + 7) = 3.8

Then, solve each equation separately for x:

1. For the first equation:
   5.2x + 7 = 3.8
   Subtract 7 from both sides:
   5.2x = 3.8 - 7
   5.2x = -3.2
   Divide both sides by 5.2:
   x = -3.2 / 5.2
   x ≈ -0.615

2. For the second equation:
   -(5.2x + 7) = 3.8
   Distribute the negative sign:
   -5.2x - 7 = 3.8
   Add 7 to both sides:
   -5.2x = 3.8 + 7
   -5.2x = 10.8
   Divide both sides by -5.2:
   x = 10.8 / -5.2
   x ≈ -2.077

So, the solutions to the equation are approximately x ≈ -0.615 and x ≈ -2.077.