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Difficulty: MediumLinear Inequalities and Absolute Value

If xx is an integer that satisfies both ∣7−2x∣≤9|7 - 2x| \le 9 and 1−x−2>2\frac{1 - x}{-2} > 2, what is the sum of all possible values of xx?

Answer: 21

Answer

The sum of all possible integer values of xx is 21.
To solve the system of inequalities, analyze each part separately. For ∣7−2x∣≤9|7 - 2x| \le 9, express it as −9≤7−2x≤9-9 \le 7 - 2x \le 9. Subtracting 77 yields −16≤−2x≤2-16 \le -2x \le 2. Dividing all parts by −2-2 requires reversing the inequality symbols, giving 8≥x≥−18 \ge x \ge -1, or −1≤x≤8-1 \le x \le 8. For 1−x−2>2\frac{1 - x}{-2} > 2, multiplying by −2-2 reverses the inequality symbol to 1−x<−41 - x < -4. Subtracting 11 gives −x<−5-x < -5, and dividing by −1-1 reverses the symbol again to x>5x > 5. The integer values satisfying both −1≤x≤8-1 \le x \le 8 and x>5x > 5 are 6,7,6, 7, and 88. Summing these integer values yields 6+7+8=216 + 7 + 8 = 21.

Step-by-Step Solution

1
Solve the absolute value inequality ∣7−2x∣≤9|7 - 2x| \le 9
-1 \le x \le 8
Unfolding the absolute value gives −9≤7−2x≤9-9 \le 7 - 2x \le 9. Subtracting 7 yields −16≤−2x≤2-16 \le -2x \le 2, and dividing by −2-2 reverses the inequalities to give −1≤x≤8-1 \le x \le 8.
2
Solve the linear inequality \frac{1 - x}{-2} > 2
x > 5
Multiplying by −2-2 flips the inequality sign to yield 1−x<−41 - x < -4. Subtracting 1 gives −x<−5-x < -5, and dividing by −1-1 flips the inequality sign again to yield x>5x > 5.
3
Find the overlapping integer values for x
x \in \{6, 7, 8\}
The integers that are greater than 5 and less than or equal to 8 are 6, 7, and 8.
4
Sum the integer solutions
21
Calculating 6 + 7 + 8 = 21.

Key Concept

Solving absolute value inequalities and compound linear inequalities with sign reversal when dividing by negative numbers.
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Topics

Linear Inequalities and Absolute Value

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