Final answer:
To solve the inequality \(|x - 4| < 11\), we consider the two cases of the absolute value, resulting in two inequalities: \(x < 15\) and \(x > -7\). Combining these gives us the solution set \(-7 < x < 15\).
Step-by-step explanation:
To solve the inequality \(|x - 4| < 11\), we need to consider what the absolute value represents. The absolute value of a number is the distance of that number from zero on the number line, without considering direction. Since the distance is less than 11, we can write two separate inequalities to represent the scenario where x - 4 is positive and where it is negative.
If x - 4 is nonnegative, we simply drop the absolute value to get x - 4 < 11. Solving for x gives us x < 15.
If x - 4 is negative, we must consider the opposite of the expression inside the absolute value, leading to the inequality -(x - 4) < 11 or -x + 4 < 11. Solving this for x, we get x > -7.
Combining these two inequalities gives us the solution set -7 < x < 15, which is the range of values for x that satisfy the original absolute value inequality.