Please solve my absolute value inequality in interval notation: $|x+6| \geq 9$
Answer (2)
To solve the inequality ∣ x + 6∣ ≥ 9 , we split it into two inequalities: x + 6 ≥ 9 and x + 6 ≤ − 9 . This results in x ≥ 3 and x ≤ − 15 , which we express in interval notation as ( − ∞ , − 15 ] ∪ [ 3 , ∞ ) .
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Split the absolute value inequality into two separate inequalities: x + 6 ≥ 9 or x + 6 ≤ − 9 .
Solve the first inequality: x + 6 ≥ 9 which gives x ≥ 3 .
Solve the second inequality: x + 6 ≤ − 9 which gives x ≤ − 15 .
Express the solution in interval notation: ( − ∞ , − 15 ] ∪ [ 3 , ∞ ) .
Explanation
Understanding the Problem We are given the absolute value inequality ∣ x + 6∣ ≥ 9 . Our goal is to solve for x and express the solution in interval notation.
Splitting into Cases An absolute value inequality of the form ∣ a x + b ∣ ≥ c is equivalent to two separate inequalities: a x + b ≥ c or a x + b ≤ − c . In our case, this means we need to solve x + 6 ≥ 9 or x + 6 ≤ − 9 .
Solving the First Inequality Let's solve the first inequality: x + 6 ≥ 9 . To isolate x , we subtract 6 from both sides of the inequality: x + 6 − 6 ≥ 9 − 6 x ≥ 3
Solving the Second Inequality Now, let's solve the second inequality: x + 6 ≤ − 9 . Again, we subtract 6 from both sides to isolate x : x + 6 − 6 ≤ − 9 − 6 x ≤ − 15
Expressing the Solution in Interval Notation The solution to the absolute value inequality is the union of the solutions to the two separate inequalities. Therefore, the solution is x ≤ − 15 or x ≥ 3 . In interval notation, this is expressed as ( − ∞ , − 15 ] ∪ [ 3 , ∞ ) .
Final Answer Therefore, the solution to the absolute value inequality ∣ x + 6∣ ≥ 9 in interval notation is ( − ∞ , − 15 ] ∪ [ 3 , ∞ ) .
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, suppose a machine is designed to fill bags with 500 grams of sugar, but the actual amount can vary by up to 15 grams. This means the amount of sugar, x , must satisfy the inequality ∣ x − 500∣ ≤ 15 . Solving this inequality helps determine the range of acceptable sugar amounts. Similarly, in engineering, absolute value inequalities can be used to specify tolerance levels for measurements or to ensure that a system operates within acceptable bounds.
