Solve the absolute value inequality.
[tex]$|x-12|-4\ \textless \ 20$[/tex]
Enter your answer in interval notation. Use "inf" for infinity and "[tex]$U$[/tex]" for union.
If there are no solutions write "no solutions".
Answer (1)
Isolate the absolute value: ∣ x − 12∣ < 24 .
Rewrite as a compound inequality: − 24 < x − 12 < 24 .
Solve for x : − 12 < x < 36 .
Express the solution in interval notation: ( − 12 , 36 ) .
Explanation
Understanding the Problem We are given the absolute value inequality ∣ x − 12∣ − 4 < 20 . Our goal is to solve for x and express the solution in interval notation.
Isolating the Absolute Value First, we isolate the absolute value term by adding 4 to both sides of the inequality: ∣ x − 12∣ − 4 + 4 < 20 + 4 ∣ x − 12∣ < 24
Rewriting as a Compound Inequality Next, we rewrite the absolute value inequality as a compound inequality: − 24 < x − 12 < 24
Solving for x Now, we solve for x by adding 12 to all parts of the inequality: − 24 + 12 < x − 12 + 12 < 24 + 12 − 12 < x < 36
Expressing the Solution in Interval Notation Finally, we express the solution in interval notation. Since x is strictly between -12 and 36, we use parentheses to denote that the endpoints are not included in the interval. The solution in interval notation is: ( − 12 , 36 )
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, consider a machine that produces bolts with a target diameter of 12 mm. Due to manufacturing variations, the actual diameter may vary slightly. If we want to ensure that the diameter is within a tolerance of 24 mm, we can express this condition as an absolute value inequality: ∣ x − 12∣ < 24 , where x is the actual diameter of the bolt. Solving this inequality gives us the range of acceptable diameters for the bolts. This concept is widely used in quality control and engineering to ensure that products meet specific standards.
