How do you solve #abs(3x-1)=14#?

Question

Eva Adamson · Accepted Answer

The solutions are #S={-13/3, 5}# #3x-1=14# #3x=14+1=15# #x=5# #-3x+1=14# #3x=-14+1=-13# #x=-13/3#

Nathan Axel · Answer

See the entire solution process below:

You must solve the term within the absolute value for both the negative and positive of what it is equivalent to providing two answers because the absolute value function takes any term, whether positive or negative, and converts it into its positive form. First Solution #3x - 1 = -14# #3x - 1 + color(red)(1) = -14 + color(red)(1)# #3x - 0 = -13# #3x = -13# #(3x)/color(red)(3) = -13/color(red)(3)# #(color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) = -13/3# #x = -13/3# Option 2) #3x - 1 = 14# #3x - 1 + color(red)(1) = 14 + color(red)(1)# #3x - 0 = 15# #3x = 15# #(3x)/color(red)(3) = 15/color(red)(3)# #(color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) = 5# #x = 5# This issue can be resolved by: #x = -13/3# and #x = 5#

Eva Adamson · Answer

#x=-13/3" or "x=5#

There are 2 solutions to equations involving the #color(blue)"absolute value"#

#rArr3x-1=color(red)(+-)14#

#color(blue)"Solution 1"#

#3x-1=14#

Add one to each side.

#3xcancel(-1)cancel(+1)=14+1#

#rArr3x=15rArrx=5#

#color(blue)"Solution 2"#

#3x-1=-14#

Add one to each side.

#3xcancel(-1)cancel(+1)=-14+1#

#rArr3x=-13rArrx=-13/3#

#color(blue)"As a check"#

#"left side "=|(3xx5)-1|=|15-1|=|14|=14color(white)(xx)✔︎#

#"left side "=|(cancel(3^1)xx-13/cancel(3)^1)-1|=|-14|=14color(white)(xx)✔︎#

#rArrx=5" or " x=-13/3" are the solutions"#

Eli Assouline · Answer

undefined To solve the equation abs(3x-1)=14, first, isolate the absolute value expression by dividing both sides by 3. This yields abs(3x-1)/3 = 14/3. Then, remove the absolute value by considering both the positive and negative cases: 3x - 1 = 14 and 3x - 1 = -14. Solve each equation separately for x.

For 3x - 1 = 14, add 1 to both sides, then divide by 3 to solve for x.
For 3x - 1 = -14, add 1 to both sides, then divide by 3 to solve for x.

The solutions for x will be the values obtained from solving these equations.