Solve the absolute value inequality: $|x+12|+5<27$

Isolate the absolute value by subtracting 5 from both sides.

To solve the inequality ∣ x + 12∣ + 5 < 27 , we isolate the absolute value to get ∣ x + 12∣ < 22 . This leads us to the compound inequality − 34 < x < 10 , which we express in interval notation as x ∈ ( − 34 , 10 ) .
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Answered by Anonymous | 2025-07-04

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 is ( − 34 , 10 ) ​ .

Explanation

Understanding the Problem We are given the absolute value inequality ∣ x + 12∣ + 5 < 27 . Our goal is to find all values of x that satisfy this inequality.

Isolating the Absolute Value First, we need to isolate the absolute value term. 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 we rewrite the absolute value inequality as a compound inequality. The 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 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 . In interval notation, this is x ∈ ( − 34 , 10 ) .


Examples
Absolute value inequalities are useful in many real-world situations. For example, suppose a machine is designed to fill bags with 500 grams of sugar, but the actual amount can vary by up to 5 grams. This means the amount of sugar, x , must satisfy the inequality ∣ x − 500∣ ≤ 5 . Solving this inequality tells us the range of possible weights: 495 ≤ x ≤ 505 . Understanding and solving absolute value inequalities helps ensure that products meet certain specifications or tolerances.

Answered by GinnyAnswer | 2025-07-04