Final answer:
To solve the inequality |3x+2| > 7, one must set up two separate inequalities: 3x + 2 < -7 and 3x + 2 > 7, then solve for 'x' in each case to find the solution set x > 5/3 or x < -3.
Step-by-step explanation:
The compound inequality that can be used to solve the inequality |3x+2| > 7 is 3x + 2 < –7 or 3x + 2 > 7.
When you encounter an absolute value inequality like |3x+2| > 7, you need to consider two scenarios because the expression inside the absolute value can be either positive or negative and still satisfy the inequality after taking the absolute value.
So you split the inequality into two parts - one for the positive scenario and one for the negative:
- For the expression within the absolute value to be positive and greater than 7, we write the inequality without the absolute value: 3x + 2 > 7.
- For the expression within the absolute value to be negative and have an absolute value greater than 7, we take the opposite of the expression and then solve: 3x + 2 < -7.
To solve these inequalities:
- To solve 3x + 2 > 7, subtract 2 from both sides, getting 3x > 5, and then divide by 3 to find x > 5/3.
- To solve 3x + 2 < -7, subtract 2 from both sides, getting 3x < -9, and then divide by 3 to find x < -3.
Therefore, the solution to the original absolute value inequality is x > 5/3 or x < -3.