solving equations with two absolute value brackets
|x+4| = |x-2|
so i want to know how to do this? do i always have to write out the four piecewise possibilities? that be ++ -+ and +-
++
x+4 = x-2
4 = -2 not solution
-+
-(x+4) = x-2
-x-4 = x-2
-4 = 2x-2
-1 = x solution
+-
x+4 = -(x-2)
x+4 = -x+2
2x+4 = 2
x = -1 solution
i forgot put the other one
--
-(x+4) = -(x-2)
-x+4 = -x+2
4 = 2 no solution
you are correct.
To solve equations with two absolute value brackets, you don't always have to write out all four possibilities. You can simplify the process by following these steps:
1. Set up two separate equations, one with the positive case, and the other with the negative case, by removing the absolute value bars.
2. For the positive case, set the expressions inside the absolute value bars equal to each other:
x + 4 = x - 2
3. Solve for x by isolating the variable on one side of the equation:
x - x = -2 - 4
0 = -6
Since -6 is not equal to zero, there is no solution for the positive case (++).
4. For the negative case, change the sign of the expression inside the absolute value bar on one side of the equation. You can choose either expression to negate. In this case, let's negate (x - 2) and set it equal to (x + 4):
-(x - 2) = x + 4
5. Distribute the negative sign to both terms inside the parentheses:
-x + 2 = x + 4
6. Solve for x by isolating the variable:
2x = 4 - 2
2x = 2
x = 1
Therefore, the negative case (-+) gives a solution of x = 1.
So, the solution to the equation |x + 4| = |x - 2| is x = -1 and x = 1.