<p><strong>953.</strong> If the set of values of \(x\) satisfying the equation \([2x] + [-2x] = \dfrac{\log_{10}(x^2 - 2x + 2) - 1}{|\log_{10}(x^2 - 2x + 2) - 1|}\) is \((a, b) - \{p_1, p_2, \ldots, p_n\}\), then find the value of \(\left(a + b + \displaystyle\sum_{i=1}^{n} p_i\right)\).<br><br>[<strong>Note:</strong> \([k]\) denotes greatest integer function less than or equal to \(k\).]</p>
Step-by-Step Solution
Key Concept: The right side of the equation equals 1 when the logarithm argument is positive (not equal to 1), and -1 when negative. The left side [2x] + [-2x] equals -1 for non-integer x and 0 for integer x. We need both sides to match simultaneously.
<p><strong>Step 1: Analyze the right-hand side (RHS).</strong></p><p>Let f(x) = log₁₀(x² - 2x + 2) - 1. The RHS equals f(x)/|f(x)|, which equals:</p><ul><li>1 if f(x) > 0, i.e., log₁₀(x² - 2x + 2) > 1</li><li>-1 if f(x) < 0, i.e., log₁₀(x² - 2x + 2) < 1</li><li>Undefined if f(x) = 0</li></ul><p><strong>Step 2: Determine when x² - 2x + 2 > 10 and when x² - 2x + 2 < 10.</strong></p><p>Note: x² - 2x + 2 = (x-1)² + 1 ≥ 1 for all x.</p><p>x² - 2x + 2 > 10 ⟹ (x-1)² > 9 ⟹ |x-1| > 3 ⟹ x < -2 or x > 4</p><p>x² - 2x + 2 < 10 ⟹ (x-1)² < 9 ⟹ |x-1| < 3 ⟹ -2 < x < 4</p><p>Thus: RHS = 1 when x ∈ (-∞, -2) ∪ (4, ∞); RHS = -1 when x ∈ (-2, 4)</p><p><strong>Step 3: Analyze the left-hand side (LHS).</strong></p><p>For any real number x, [2x] + [-2x] = -1 if x ∉ ℤ, and [2x] + [-2x] = 0 if x ∈ ℤ.</p><p>Proof: If x = n + f where n ∈ ℤ and 0 < f < 1, then [2x] = [2n + 2f] = 2n + 1 and [-2x] = [-2n - 2f] = -2n - 2, so sum = -1.</p><p><strong>Step 4: Match LHS = RHS.</strong></p><p>Case 1: LHS = 1. This is impossible since LHS ∈ {-1, 0}.</p><p>Case 2: LHS = -1. This requires x ∉ ℤ AND x ∈ (-2, 4).</p><p>Therefore: x ∈ (-2, 4) - {integers in (-2, 4)} = (-2, 4) - {-1, 0, 1, 2, 3}</p><p><strong>Step 5: Identify a, b, and the excluded points.</strong></p><p>From the solution set (-2, 4) - {-1, 0, 1, 2, 3}:</p><ul><li>a = -2</li><li>b = 4</li><li>Excluded points: p₁ = -1, p₂ = 0, p₃ = 1, p₄ = 2, p₅ = 3</li><li>n = 5</li></ul><p><strong>Step 6: Calculate a + b + Σpᵢ.</strong></p><p>a + b + Σpᵢ = -2 + 4 + (-1 + 0 + 1 + 2 + 3) = 2 + 5 = 7</p><p><strong>∴ Answer: 7</strong></p>
Correct Answer: 7