Solve the absolute value inequality. Other than $\varnothing$, use interval notation to express the solution set and graph the solution set on a number line.
$|x-6|+8 \geq 17$
Answer (2)
Isolate the absolute value: ∣ x − 6∣ g e q 9 .
Split into two cases: x − 6 g e q 9 and x − 6 ≤ − 9 .
Solve each case: xg e q 15 or x ≤ − 3 .
Express the solution in interval notation: ( − ∞ , − 3 ] ∪ [ 15 , ∞ ) .
Explanation
Understanding the Inequality We are given the absolute value inequality ∣ x − 6∣ + 8 ≥ 17 . 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 subtracting 8 from both sides of the inequality: ∣ x − 6∣ + 8 − 8 ≥ 17 − 8 which simplifies to ∣ x − 6∣ ≥ 9
Solving the Two Cases Now, we consider two cases based on the definition of absolute value. Case 1: The expression inside the absolute value is non-negative, i.e., x − 6 ≥ 0 . In this case, ∣ x − 6∣ = x − 6 , so we have x − 6 ≥ 9 Adding 6 to both sides gives x ≥ 15 Case 2: The expression inside the absolute value is negative, i.e., x − 6 < 0 . In this case, ∣ x − 6∣ = − ( x − 6 ) , so we have − ( x − 6 ) ≥ 9 Multiplying both sides by -1 (and flipping the inequality sign) gives x − 6 ≤ − 9 Adding 6 to both sides gives x ≤ − 3
Expressing the Solution in Interval Notation Combining the two cases, we have x ≥ 15 or x ≤ − 3 . In interval notation, this is written as ( − ∞ , − 3 ] ∪ [ 15 , ∞ )
Final Answer The solution set is ( − ∞ , − 3 ] ∪ [ 15 , ∞ ) . This means that x can be any number less than or equal to -3, or any number greater than or equal to 15.
Examples
Absolute value inequalities can be used to model real-world situations where a quantity must be within a certain range of a target value. For example, a machine that produces parts needs to create pieces that are within a certain tolerance of a specified measurement. If the specified measurement is 6 units, and the tolerance is 9 units, then the actual measurement, x , must satisfy ∣ x − 6∣ ≤ 9 . This means the part can be at most 15 units or at least -3 units. Understanding and solving absolute value inequalities helps ensure that the parts produced meet the required specifications.
The solution to the inequality ∣ x − 6∣ + 8 ≥ 17 is represented by the set ( − ∞ , − 3 ] ∪ [ 15 , ∞ ) . This demonstrates that values of x can be either less than or equal to -3 or greater than or equal to 15. Therefore, the solution includes all x in those intervals.
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