Examkin

Question

Difficulty: HardIntegers, Absolute Value, and Number Lines

On a standard number line, the coordinate of point PP is an integer xx. If the sum of the distances from PP to −2-2 and from PP to 44 is equal to the distance from PP to 1010, what is the sum of all possible values of xx?

Answer: -4

Answer

The sum of all possible values of the integer xx is −4-4.
The correct answer is −4-4. The distance between any two coordinates aa and bb on a number line is defined as ∣a−b∣|a - b|. Translating the problem gives the equation ∣x+2∣+∣x−4∣=∣x−10∣|x + 2| + |x - 4| = |x - 10|. Breaking the number line into intervals around the critical points x=−2x = -2, x=4x = 4, and x=10x = 10 yields two valid integer solutions: x=−8x = -8 and x=4x = 4. The sum of these values is −8+4=−4-8 + 4 = -4.

Step-by-Step Solution

1
Express the distances on the number line using absolute values.
The distance from P(x)P(x) to −2-2 is ∣x−(−2)∣=∣x+2∣|x - (-2)| = |x + 2|. The distance from P(x)P(x) to 44 is ∣x−4∣|x - 4|. The distance from P(x)P(x) to 1010 is ∣x−10∣|x - 10|. The equation is ∣x+2∣+∣x−4∣=∣x−10∣|x + 2| + |x - 4| = |x - 10|.
The absolute value ∣a−b∣|a - b| represents the distance between points aa and bb on a standard number line.
2
Solve the equation by testing the intervals defined by the critical points x=−2x = -2, x=4x = 4, and x=10x = 10.
We analyze the four intervals:
- For x<−2x < -2: −(x+2)−(x−4)=−(x−10)  ⟹  −2x+2=−x+10  ⟹  x=−8-(x + 2) - (x - 4) = -(x - 10) \implies -2x + 2 = -x + 10 \implies x = -8. Since −8<−2-8 < -2, this is a valid solution.
- For −2≤x<4-2 \leq x < 4: (x+2)−(x−4)=−(x−10)  ⟹  6=−x+10  ⟹  x=4(x + 2) - (x - 4) = -(x - 10) \implies 6 = -x + 10 \implies x = 4. Since 44 is not in [−2,4)[-2, 4), there is no solution in this interval.
- For 4≤x<104 \leq x < 10: (x+2)+(x−4)=−(x−10)  ⟹  2x−2=−x+10  ⟹  3x=12  ⟹  x=4(x + 2) + (x - 4) = -(x - 10) \implies 2x - 2 = -x + 10 \implies 3x = 12 \implies x = 4. Since 44 is in [4,10)[4, 10), this is a valid solution.
- For x≥10x \geq 10: (x+2)+(x−4)=x−10  ⟹  2x−2=x−10  ⟹  x=−8(x + 2) + (x - 4) = x - 10 \implies 2x - 2 = x - 10 \implies x = -8. Since −8<10-8 < 10, there is no solution in this interval.
Absolute value terms change sign at their critical points, requiring case-by-case evaluation.
3
Sum all valid integer solutions.
The valid values for xx are −8-8 and 44. Their sum is −8+4=−4-8 + 4 = -4.
The problem asks for the sum of all possible values of xx.

Key Concept

Representing geometric distances on a number line using absolute value equations and solving them using interval analysis.
Rate this question

Topics

Integers, Absolute Value, and Number Lines

More questions from this topic