Question

1.Solve the equation. Check for extraneous solutions.

Answer 1

Okay so I did the math. 

1. Two solutions were found : 
z=-1
z=2/3
Step  1  :Rearrange this Absolute Value Equation

Absolute value equalitiy entered
      |2z-3| = 4z-1 

Step  2  :Clear the Absolute Value Bars

Clear the absolute-value bars by splitting the equation into its two cases, one for the Positive case and the other for the Negative case.

The Absolute Value term is |2z-3|

 For the Negative case we'll use -(2z-3) 

For the Positive case we'll use (2z-3) 

Step  3  :Solve the Negative Case

      -(2z-3) = 4z-1 

     Multiply
      -2z+3 = 4z-1 

     Rearrange and Add up
      -6z = -4 

     Divide both sides by 6 
      -z = -(2/3) 

     Multiply both sides by (-1) 
      z = (2/3) 
     Which is the solution for the Negative Case

Step  4  :Solve the Positive Case

      (2z-3) = 4z-1 

     Rearrange and Add up
      -2z = 2 

     Divide both sides by 2 
      -z = 1 

     Multiply both sides by (-1) 
      z = -1 
     Which is the solution for the Positive Case
Step  5  :Wrap up the solution

 z=2/3
 z=-1

2. -6 < x < 26/3

Step  1  :Rearrange this Absolute Value Inequality

Absolute value inequalitiy entered
      |3x-4|+5 < 27 

Another term is moved / added to the right hand side.

      |3x-4| < 22 

Step  2  :Clear the Absolute Value Bars

Clear the absolute-value bars by splitting the equation into its two cases, one for the Positive case and the other for the Negative case.

The Absolute Value term is |3x-4|

 For the Negative case we'll use -(3x-4) 

For the Positive case we'll use (3x-4) 

Step  3  :Solve the Negative Case

      -(3x-4) < 22 

     Multiply
      -3x+4 < 22 

     Rearrange and Add up
      -3x < 18 

     Divide both sides by 3 
      -x < 6 

     Multiply both sides by (-1) 
     Remember to flip the inequality sign 
      x > -6 
     Which is the solution for the Negative Case

Step  4  :Solve the Positive Case

      (3x-4) < 22 

     Rearrange and Add up
      3x < 26 

     Divide both sides by 3 
      x < (26/3) 

     Which is the solution for the Positive Case

Step  5  :Wrap up the solution

    -6 < x < 26/3

Solution in Interval Notation

    (-6,26/3) 

HOPE THIS HELPS :D