Solve the absolute value inequality: [tex]$|x+12|+5<27$[/tex]
Isolate the absolute value by subtracting 5 from both sides.
[tex]$|x+12|<22$[/tex]
Separate into a compound inequality.
A. [tex]$-22 x+12<22$[/tex]
Answer (2)
The solution to the absolute value inequality ∣ x + 12∣ + 5 < 27 is found by isolating the absolute value to get − 22 < x < 10 . In interval notation, this means x in ( − 34 , 10 ) . Thus, the correct multiple choice option is A: − 22 < x + 12 < 22 .
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Isolate the absolute value: ∣ x + 12∣ < 22 .
Rewrite as a compound inequality: − 22 < x + 12 < 22 .
Solve for x by subtracting 12: − 34 < x < 10 .
The solution is x ∈ ( − 34 , 10 ) , which means − 34 < x < 10 . − 34 < x < 10
Explanation
Isolating the Absolute Value 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. Let's start by isolating the absolute value term.
Simplifying the Inequality Subtract 5 from both sides of the inequality: ∣ x + 12∣ + 5 − 5 < 27 − 5 ∣ x + 12∣ < 22
Creating a Compound Inequality Now, we rewrite the absolute value 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 expressed as: − 22 < x + 12 < 22
Solving for x To solve for x , we subtract 12 from all parts of the compound inequality: − 22 − 12 < x + 12 − 12 < 22 − 12 − 34 < x < 10
Final Answer The solution to the absolute value inequality is − 34 < x < 10 . This means that x can be any value between -34 and 10, not including -34 and 10. In interval notation, this is written as x ∈ ( − 34 , 10 ) .
Examples
Absolute value inequalities are useful in various real-world scenarios. For instance, consider a manufacturing process where a machine is designed to produce bolts with a length of 5 cm. However, due to slight variations in the manufacturing process, the actual length of the bolts can deviate by a certain tolerance, say 0.2 cm. This situation can be modeled using an absolute value inequality: ∣ L − 5∣ < 0.2 , where L represents the actual length of the bolt. Solving this inequality will give the range of acceptable bolt lengths, ensuring that the bolts meet the required specifications. In this case, the solution would be 4.8 < L < 5.2 , meaning the bolt length must be between 4.8 cm and 5.2 cm to be considered acceptable.
