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Difficulty: HardReal Numbers, Number Line, and Absolute Value

On the real number line, xx is a real number such that its distance from 11 is at most 66, and its distance from −2-2 is at least 44. Which of the following inequalities represents all possible values of xx?

  1. 2≤x≤72 \le x \le 7Answer
  2. B
    −5≤x≤7-5 \le x \le 7
  3. C
    −6≤x≤7-6 \le x \le 7
  4. D
    −5≤x≤2-5 \le x \le 2
  5. E
    −5≤x≤−2 or 2≤x≤7-5 \le x \le -2 \text{ or } 2 \le x \le 7

Answer

The inequality 2≤x≤72 \le x \le 7 represents all possible values of xx.
The distance condition 'at most 6 units from 1' translates to ∣x−1∣≤6|x - 1| \le 6, yielding the interval [−5,7][-5, 7]. The condition 'at least 4 units from -2' translates to ∣x+2∣≥4|x + 2| \ge 4, yielding x≤−6x \le -6 or x≥2x \ge 2. Finding the values that belong to both conditions requires taking the intersection of [−5,7][-5, 7] and (−∞,−6]∪[2,∞)(-\infty, -6] \cup [2, \infty). Since [−5,7][-5, 7] does not overlap with (−∞,−6](-\infty, -6], the only overlapping region is [2,7][2, 7], which corresponds to 2≤x≤72 \le x \le 7.

Step-by-Step Solution

1
Express the geometric distance conditions as absolute value inequalities
Condition 1: ∣x−1∣≤6|x - 1| \le 6. Condition 2: ∣x−(−2)∣=∣x+2∣≥4|x - (-2)| = |x + 2| \ge 4.
The distance between two numbers aa and bb on the number line is given by ∣a−b∣|a - b|.
2
Solve the first inequality ∣x−1∣≤6|x - 1| \le 6
−6≤x−1≤6  ⟹  −5≤x≤7-6 \le x - 1 \le 6 \implies -5 \le x \le 7.
An absolute value inequality of the form ∣u∣≤k|u| \le k (for k≥0k \ge 0) unwraps to −k≤u≤k-k \le u \le k.
3
Solve the second inequality ∣x+2∣≥4|x + 2| \ge 4
x+2≥4x + 2 \ge 4 or x+2≤−4  ⟹  x≥2x + 2 \le -4 \implies x \ge 2 or x≤−6x \le -6.
An absolute value inequality of the form ∣u∣≥k|u| \ge k (for k≥0k \ge 0) unwraps to u≥ku \ge k or u≤−ku \le -k.
4
Find the intersection of the two solution sets
Combine [−5,7][-5, 7] with (−∞,−6]∪[2,∞)(-\infty, -6] \cup [2, \infty). The intersection of [−5,7][-5, 7] and (−∞,−6](-\infty, -6] is empty since −5>−6-5 > -6. The intersection of [−5,7][-5, 7] and [2,∞)[2, \infty) is [2,7][2, 7], which means 2≤x≤72 \le x \le 7.
The variable xx must satisfy both conditions simultaneously.

Key Concept

Distance on a number line expressed as absolute value inequalities and finding overlapping intervals
Estimated Time:2m 0s
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Topics

Real Numbers, Number Line, and Absolute Value

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