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 or x − 6 ≤ − 9 .
Solve each case: xg e q 15 or x ≤ − 3 .
Express the solution in interval notation: ( − ∞ , − 3 ] ∪ [ 15 , ∞ ) .
Explanation
Understanding the Problem We are asked to solve the absolute value inequality ∣ x − 6∣ + 8 ≥ 17 , express the solution set in interval notation, and graph the solution set on a number line.
Isolating the Absolute Value First, we isolate the absolute value term by subtracting 8 from both sides of the inequality: ∣ x − 6∣ ≥ 17 − 8 ∣ x − 6∣ ≥ 9
Considering Two Cases Now, we consider the two cases for the absolute value inequality: Case 1: x − 6 ≥ 9 Case 2: x − 6 ≤ − 9
Solving Case 1 For Case 1, we solve the inequality by adding 6 to both sides: x ≥ 9 + 6 x ≥ 15
Solving Case 2 For Case 2, we solve the inequality by adding 6 to both sides: x ≤ − 9 + 6 x ≤ − 3
Expressing the Solution in Interval Notation The solution set is the union of the solutions from both cases. In interval notation, this is: ( − ∞ , − 3 ] ∪ [ 15 , ∞ )
Final Answer The solution set includes all real numbers less than or equal to -3, and all real numbers greater than or equal to 15.
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, consider a machine that fills bags with coffee. The target weight is 16 ounces, but the machine isn't perfect. The absolute value inequality ∣ w − 16∣ ≤ 0.5 describes the weights w that are within 0.5 ounces of the target. Solving this inequality helps determine the range of acceptable weights for the coffee bags, ensuring quality control.
To solve the inequality ∣ x − 6∣ + 8 ≥ 17 , isolate the absolute value to get ∣ x − 6∣ ≥ 9 . This gives two cases: x ≥ 15 or x ≤ − 3 , resulting in the solution set ( − ∞ , − 3 ] ∪ [ 15 , ∞ ) .
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