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Difficulty: MediumIntegers, Absolute Value, and Number Lines

A chemical mixture is kept in a temperature-controlled chamber where the temperature TT, in degrees Celsius, is restricted to the range −10≤T≤10-10 \le T \le 10. The mixture remains stable as long as the temperature satisfies the inequality:

2−3∣T−2∣≥−102 - 3|T - 2| \ge -10

What is the sum of all integer values of TT in this range for which the mixture is stable?

  1. A
    19
  2. B
    -14
  3. 18Answer
  4. D
    -18
  5. E
    20

Answer

18
The correct answer is 18. Subtracting 2 from both sides of the inequality 2−3∣T−2∣≥−102 - 3|T - 2| \ge -10 gives −3∣T−2∣≥−12-3|T - 2| \ge -12. Dividing by −3-3 and reversing the inequality sign yields ∣T−2∣≤4|T - 2| \le 4, which expands to the compound inequality −4≤T−2≤4-4 \le T - 2 \le 4. Adding 2 to all parts gives −2≤T≤6-2 \le T \le 6. All integers in this range (−2,−1,0,1,2,3,4,5,6-2, -1, 0, 1, 2, 3, 4, 5, 6) are within the chamber's allowed range of [−10,10][-10, 10]. Their sum is (−2)+(−1)+0+1+2+3+4+5+6=18(-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 = 18.

Step-by-Step Solution

1
Isolate the absolute value expression by subtracting 2 from both sides.
−3∣T−2∣≥−12-3|T - 2| \ge -12
To solve an absolute value inequality, we must first isolate the absolute value term on one side.
2
Divide both sides by −3-3 and reverse the inequality sign.
∣T−2∣≤4|T - 2| \le 4
Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.
3
Rewrite the absolute value inequality as a compound inequality.
−4≤T−2≤4-4 \le T - 2 \le 4
An inequality of the form ∣x∣≤a|x| \le a (where a≥0a \ge 0) is equivalent to −a≤x≤a-a \le x \le a.
4
Solve for TT by adding 2 to all three parts of the inequality.
−2≤T≤6-2 \le T \le 6
Adding 2 isolates the variable TT to find the range of stable temperatures.
5
Identify all integer values of TT in the stable range and calculate their sum.
Sum = 18
The integers in the interval [−2,6][-2, 6] are −2,−1,0,1,2,3,4,5,6-2, -1, 0, 1, 2, 3, 4, 5, 6. Since the chamber is restricted to [−10,10][-10, 10], all of these values are valid. Summing them: (−2)+(−1)+0+1+2+3+4+5+6=18(-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 = 18.

Key Concept

Solving absolute value inequalities and performing operations with signed integers.
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Topics

Integers, Absolute Value, and Number Lines

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