Solve the absolute value inequality: $|x+12|+5<27$
Isolate the absolute value by subtracting 5 from both sides.
$|x+12|<22$
Answer (2)
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Isolate the absolute value: ∣ x + 12∣ < 22 .
Rewrite as a compound inequality: − 22 < x + 12 < 22 .
Solve for x : − 34 < x < 10 .
Express the solution in interval notation: x ∈ ( − 34 , 10 ) . The solution to the absolute value inequality is x ∈ ( − 34 , 10 ) .
Explanation
Problem Analysis We are given the absolute value inequality ∣ x + 12∣ + 5 < 27 . Our goal is to isolate x and find the range of values that satisfy this inequality.
Isolating the Absolute Value First, we need to isolate the absolute value. To do this, we subtract 5 from both sides of the inequality: ∣ x + 12∣ + 5 − 5 < 27 − 5 ∣ x + 12∣ < 22
Rewriting as a Compound Inequality Now that we have isolated the absolute value, we can rewrite the inequality as a compound inequality. The absolute value inequality ∣ x + 12∣ < 22 means that the distance between x + 12 and 0 is less than 22. This can be written as: − 22 < x + 12 < 22
Solving for x To solve for x , we subtract 12 from all parts of the inequality: − 22 − 12 < x + 12 − 12 < 22 − 12 − 34 < x < 10
Expressing the Solution in Interval Notation This means that x must be greater than -34 and less than 10. In interval notation, the solution is: x ∈ ( − 34 , 10 )
Final Answer Therefore, the solution to the absolute value inequality ∣ x + 12∣ + 5 < 27 is − 34 < x < 10 .
Examples
Absolute value inequalities are useful in various real-world scenarios. For example, consider a manufacturing process where a machine is set to produce parts with a specific length. Due to slight variations in the machine's operation, the actual length of the parts may deviate from the target length. An absolute value inequality can be used to define the acceptable range of deviation. If the target length is L and the acceptable deviation is d , then the actual length x of the parts must satisfy the inequality ∣ x − L ∣ < d . This ensures that all manufactured parts fall within the specified tolerance range, maintaining quality control.
