Solve: $|2x + 8| < 2$. Give your answer as an interval.
Answer (1)
Rewrite the absolute value inequality as a compound inequality: − 2 < 2 x + 8 < 2 .
Subtract 8 from all parts: − 10 < 2 x < − 6 .
Divide all parts by 2: − 5 < x < − 3 .
Express the solution as an interval: ( − 5 , − 3 ) .
Explanation
Understanding the Problem We are given the absolute value inequality ∣2 x + 8∣ < 2 . Our goal is to solve for x and express the solution as an interval.
Rewriting the Inequality To solve an absolute value inequality of the form ∣ a x + b ∣ < c , we rewrite it as a compound inequality without the absolute value: − c < a x + b < c . In our case, this means: − 2 < 2 x + 8 < 2
Isolating x Now, we want to isolate x in the middle. First, subtract 8 from all parts of the inequality: − 2 − 8 < 2 x + 8 − 8 < 2 − 8 − 10 < 2 x < − 6
Solving for x Next, divide all parts of the inequality by 2: 2 − 10 < 2 2 x < 2 − 6 − 5 < x < − 3
Expressing the Solution as an Interval The solution to the inequality is all x such that − 5 < x < − 3 . In interval notation, this is written as ( − 5 , − 3 ) .
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, in manufacturing, you might want to ensure that the dimensions of a product are within a certain tolerance of the specified size. If the specified size is x and the tolerance is t , then the actual size a must satisfy ∣ a − x ∣ < t . This ensures that the product meets the required quality standards. Similarly, in finance, you might use absolute value inequalities to model the risk associated with an investment. If you want to keep the return on investment within a certain range, you can use an absolute value inequality to represent this constraint.
