Question

Solve the equation. Check for extraneous solutions. |x-2|=4x+4

Answer 1

| x - 2 | = 4x + 4 is our given equation. In an absolute value equation, we solve the original expression as our first equation. Our second one is that we multiply the right side by -1 (using the definition of absolute value)

Case 1: original equation

|x - 2| = 4x + 4          <-----original

x - 2 = 4x + 4           <-----take the positive one

-2 = 3x + 4               <-----subtract x on both sides

-6 = 3x                     <-----subtract 4 on both sides

-2 = x                       <-----divide both sides by -3

Case 2: Opposite equation

| x - 2 | = 4x + 4        <------original equation

x - 2 = -(4x + 4)         <------take the negative of the right side

x - 2 = -4x -4             <----multiply the right side by -1

-2 = -5x - 4               <------subtract x on both sides

2 = -5x                     <-------add 4 on both sides

x = -2/5                    <--------divide both sides by -5

Now we have two solutions. We need to check for extraneous solutions because of all the manipulations.

Check:

| x - 2 | = 4x + 4          <------- original

| -2 -2 | = 4(-2) + 4      <--------use x = -2

| -4 | = -8 + 4

4 = -4                    Not a solution

| x - 2 | = 4x + 4          <------- original

| -2/5 - 2 | = 4(-2/5) + 4    <----use x = -2/5

|-2/5 - 2 | = -8/5 + 4

|-2/5 - 10/5 | = -8/5 + 20/5

|-12/5 | = 12/5

12/5 = 12/5            Solution

Therefore x = -2/5