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Difficulty: Very hardInequalities and Absolute Value Equations

How many integer values of xx satisfy the inequality ∣2x−7∣−3∣x+1∣−4≤0\frac{|2x - 7| - 3}{|x + 1| - 4} \leq 0?

  1. A
    7
  2. B
    8
  3. 9Answer
  4. D
    10
  5. E
    11

Answer

9 integer values satisfy the given inequality.
The quotient of two algebraic expressions is non-positive (≤0\leq 0) when the numerator is zero (and denominator non-zero) or when the numerator and denominator have opposite signs. Testing these cases yields two disjoint intervals: −5<x≤2-5 < x \leq 2 and 3<x≤53 < x \leq 5. Within these intervals, the integers −4,−3,−2,−1,0,1,2,4,-4, -3, -2, -1, 0, 1, 2, 4, and 55 satisfy the inequality, yielding exactly 9 integer solutions.

Step-by-Step Solution

1
Determine where the numerator and denominator equal zero to set critical points and domain restrictions.
Numerator is zero at ∣2x−7∣=3  ⟹  x=2|2x - 7| = 3 \implies x = 2 or x=5x = 5. Denominator is zero at ∣x+1∣=4  ⟹  x=−5|x + 1| = 4 \implies x = -5 or x=3x = 3, which are excluded from the domain.
Division by zero is undefined, so x≠−5x \neq -5 and x≠3x \neq 3.
2
Analyze Case 1: Numerator ≥0\geq 0 and Denominator <0< 0.
Numerator ≥0  ⟹  x≤2\geq 0 \implies x \leq 2 or x≥5x \geq 5. Denominator <0  ⟹  −5<x<3< 0 \implies -5 < x < 3. Intersecting these gives −5<x≤2-5 < x \leq 2.
A fraction is non-positive when the numerator and denominator have opposite signs.
3
Analyze Case 2: Numerator ≤0\leq 0 and Denominator >0> 0.
Numerator ≤0  ⟹  2≤x≤5\leq 0 \implies 2 \leq x \leq 5. Denominator >0  ⟹  x<−5> 0 \implies x < -5 or x>3x > 3. Intersecting these gives 3<x≤53 < x \leq 5.
A fraction is non-positive when the numerator and denominator have opposite signs.
4
Combine solution intervals and count integer solutions.
Combined solution set is (−5,2]∪(3,5](-5, 2] \cup (3, 5]. The integer values in this set are −4,−3,−2,−1,0,1,2,4,5-4, -3, -2, -1, 0, 1, 2, 4, 5, giving a total count of 9.
Integrate all valid cases while excluding values that make the denominator zero.

Key Concept

Solving Rational Inequalities with Absolute Values
Estimated Time:2m 30s
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Topics

Inequalities and Absolute Value Equations

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