Final answer:
The solution to the inequality -5 < |2x+7| < 3 is obtained by splitting the inequality into two cases based on the expression inside the absolute value. These cases are evaluated separately to find the solution set -5 < x < -2.
Step-by-step explanation:
For the inequality -5 < |2x+7| < 3, we need to evaluate the absolute value expression under two conditions: when the expression inside is positive and when it's negative. Splitting the inequality for the case when the expression inside absolute value (2x+7) is positive, we get:
Case 1: 2x+7 < 3, which simplifies to 2x < -4, hence x < -2.
Case 2: -(2x+7) < 3, which simplifies to -2x - 7 < 3, and further to -2x < 10, hence x > -5.
Now, combining these two cases, we get the solution set of x > -5 and x < -2, meaning -5 < x < -2.
This method extends to solving other similar absolute value inequalities by considering both cases for the expression inside the absolute value. Remember to reverse the inequality when multiplying or dividing by a negative number.