|(3x+2)/2|=|(2x-3)/3|
To solve the given equation, you can follow the steps below:
Step 1: Eliminate the absolute value signs.
The absolute value signs in the equation indicate that the expression inside them can be positive or negative. To eliminate the absolute value signs, you need to consider both possibilities.
For positive case:
(3x + 2)/2 = (2x - 3)/3
For negative case:
-(3x + 2)/2 = (2x - 3)/3
Step 2: Solve for x in each case.
For the positive case:
Multiply both sides of the equation by 2 and 3 to eliminate the denominators:
3(3x + 2) = 2(2x - 3)
Expand the equation:
9x + 6 = 4x - 6
Move the variable terms to one side and the constant terms to the other:
9x - 4x = -6 - 6
5x = -12
Divide both sides by 5 to isolate x:
x = -12/5 or x = -2.4
For the negative case:
Multiply both sides of the equation by 2 and -3 to eliminate the denominators:
-2(3x + 2) = 3(2x - 3)
Expand the equation:
-6x - 4 = 6x - 9
Move the variable terms to one side and the constant terms to the other:
-6x - 6x = -9 + 4
-12x = -5
Divide both sides by -12 to isolate x:
x = -5/-12 or x = 5/12
Step 3: Check for extraneous solutions.
Since we squared the equation in order to remove the absolute value signs, it's important to check if any of the solutions satisfy the original equation.
For x = -2.4:
LHS = |(3(-2.4) + 2)/2| = |-4.2/2| = |-2.1| = 2.1
RHS = |(2(-2.4) - 3)/3| = |-7.8/3| = |-2.6| = 2.6
Since the LHS is not equal to the RHS, x = -2.4 is not a valid solution.
For x = 5/12:
LHS = |(3(5/12) + 2)/2| = |(5/4 + 2)/2| = |(13/4)/2| = |(13/8)| = 13/8
RHS = |(2(5/12) - 3)/3| = |(5/6 - 3)/3| = |-13/6| = 13/6
Since the LHS is equal to the RHS, x = 5/12 is a valid solution.
Therefore, the equation has one valid solution: x = 5/12.