| 3x+2 | + | 3x | =7

To solve the equation |3x + 2| + |3x| = 7, you'll need to consider four possible cases based on the absolute value expressions. The absolute value of a number is either the number itself if it's positive or the negation of the number if it's negative. So, let's break it down: Case 1: 3x + 2 is positive, and 3x is positive. Case 2: 3x + 2 is positive, and 3x is negative. Case 3: 3x + 2 is negative, and 3x is positive. Case 4: 3x + 2 is negative, and 3x is negative. Let's solve each case separately: Case 1: 3x + 2 is positive, and 3x is positive: In this case, you have two positive expressions inside the absolute values: (3x + 2) + (3x) = 7 Combine like terms: 6x + 2 = 7 Subtract 2 from both sides: 6x = 7 - 2 6x = 5 Divide both sides by 6: x = 5/6 Case 2: 3x + 2 is positive, and 3x is negative: In this case, you have a positive expression and a negative expression inside the absolute values: (3x + 2) - (3x) = 7 Combine like terms: 2 = 7 This equation is not possible because it leads to a contradiction. There are no solutions in this case. Case 3: 3x + 2 is negative, and 3x is positive: In this case, you have a negative expression and a positive expression inside the absolute values: -(3x + 2) + (3x) = 7 Combine like terms: -2 = 7 This equation is also not possible because it leads to a contradiction. There are no solutions in this case. Case 4: 3x + 2 is negative, and 3x is negative: In this case, you have two negative expressions inside the absolute values: -(3x + 2) - (3x) = 7 Combine like terms: -6x - 2 = 7 Add 2 to both sides: -6x = 7 + 2 -6x = 9 Divide both sides by -6: x = 9 / (-6) x = -3/2 So, the solutions to the original equation are: x = 5/6 (from Case 1) x = -3/2 (from Case 4)