What are the solutions to the absolute value inequality $|x-70| \leq 3$?
Remember, the inequality can be written as $-3 \leq x-70 \leq 3$ or as $x-70 \leq 3$ and $x-70 \geq -3$.
A. $x \leq 67$
B. $x \leq 73$
C. $67 \leq x \leq 73$
D. $67 \geq x \geq 73$
Answer (2)
The solutions to the inequality ∣ x − 70∣ ≤ 3 can be found by rewriting it as the compound inequality − 3 ≤ x − 70 ≤ 3 . After isolating x , we find 67 ≤ x ≤ 73 . Therefore, the correct answer is option C: 67 ≤ x ≤ 73 .
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Rewrite the absolute value inequality as a compound inequality: − 3 ≤ x − 70 ≤ 3 .
Add 70 to all parts of the inequality to isolate x : 67 ≤ x ≤ 73 .
The solution is all x such that 67 ≤ x ≤ 73 .
The solutions to the absolute value inequality are 67 ≤ x ≤ 73 .
Explanation
Understanding the Problem We are given the absolute value inequality ∣ x − 70∣ ≤ 3 . Our goal is to find all values of x that satisfy this inequality.
Rewriting the Inequality An absolute value inequality of the form ∣ a ∣ ≤ b can be rewritten as a compound inequality: − b ≤ a ≤ b . Applying this to our problem, we rewrite the given inequality as: − 3 ≤ x − 70 ≤ 3
Isolating x To isolate x , we add 70 to all parts of the inequality: − 3 + 70 ≤ x − 70 + 70 ≤ 3 + 70
Simplifying Simplifying the inequality, we get: 67 ≤ x ≤ 73
Final Answer This means that x must be greater than or equal to 67 and less than or equal to 73. Therefore, the solution to the absolute value inequality is all x in the interval [ 67 , 73 ] .
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, suppose a machine is designed to fill bags with 70 grams of sugar, but the actual amount can vary by up to 3 grams. The inequality ∣ x − 70∣ ≤ 3 describes the possible weights, x , of the filled bags. This ensures that the bags are filled within an acceptable range around the target weight. Understanding and solving such inequalities helps in quality control and maintaining consistency in manufacturing processes.
