Solve: $|x+4|>7$
Give your answer as an interval.
Answer (1)
Split the absolute value inequality into two cases: 7"> x + 4 > 7 and x + 4 < − 7 .
Solve the first case: 7 Arr x > 3"> x + 4 > 7 A rr x > 3 .
Solve the second case: x + 4 < − 7 A rr x < − 11 .
Express the solution as the union of the two intervals: ( − ∞ , − 11 ) ∪ ( 3 , ∞ ) . The final answer is ( − ∞ , − 11 ) ∪ ( 3 , ∞ )
Explanation
Understanding the Problem We are given the absolute value inequality 7"> ∣ x + 4∣ > 7 . Our goal is to find all values of x that satisfy this inequality and express the solution as an interval.
Splitting into Cases The absolute value inequality 7"> ∣ x + 4∣ > 7 means that the distance between x + 4 and 0 is greater than 7. This leads to two separate cases: either x + 4 is greater than 7, or x + 4 is less than -7.
Solving Case 1 Case 1: 7"> x + 4 > 7 . To solve this inequality, we subtract 4 from both sides: 7-4"> x + 4 − 4 > 7 − 4 3"> x > 3
Solving Case 2 Case 2: x + 4 < − 7 . To solve this inequality, we subtract 4 from both sides: x + 4 − 4 < − 7 − 4 x < − 11
Combining the Solutions The solution to the inequality 7"> ∣ x + 4∣ > 7 is the union of the solutions from the two cases. In interval notation, 3"> x > 3 is represented as ( 3 , ∞ ) , and x < − 11 is represented as ( − ∞ , − 11 ) . Therefore, the solution to the inequality is the union of these two intervals: ( − ∞ , − 11 ) ∪ ( 3 , ∞ ) .
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, suppose a machine is designed to fill bags with 5 pounds of sugar, but the actual weight can vary by up to 0.2 pounds. This means the weight w of the sugar in a bag must satisfy ∣ w − 5∣ ≤ 0.2 . Solving this inequality helps determine the range of acceptable weights for the bags of sugar. Similarly, in engineering, absolute value inequalities are used to specify tolerances in manufacturing processes, ensuring that components meet certain specifications within acceptable limits.
