Zad. 30 a) 3·(2x - 1) + 5x = 2 - [4x - (1 - 2x)] 6x - 3 + 5x = 2 - (4x - 1 + 2x) 11x - 3 = 2 - (6x - 1) 11x - 3 = 2 - 6x + 1 11x + 6x = 3 + 3 17x = 6 /:17 [latex]x = frac{6}{17}[/latex] b) [latex]frac{x+3}{2} + frac{4x}{3} = frac{2x-1}{6} + 2 / cdot 6[/latex] [latex]3 cdot (x+3) + 2 cdot 4x = 2x-1+ 12 [/latex] [latex]3x+9 + 8x = 2x + 11 [/latex] [latex]11x - 2x = 11 - 9[/latex] [latex]9x = 2 / : 9[/latex] [latex]x = frac{2}{9}[/latex] c) [latex]frac{5x+2}{5} + x = frac{3x-4}{2} + frac{1}{2} x / cdot 10[/latex] [latex]2 cdot (5x + 2) + 10x = 5 cdot (3x - 4) + 5x[/latex] [latex]10x + 4 + 10x = 15x - 20 + 5x [/latex] [latex]20x + 4 = 20x - 20 [/latex] [latex]20x -20x = - 20 -4[/latex] [latex]0 = - 24[/latex] sprzeczność [latex]x in emptyset [/latex] d) (x + 4)² + 14 = (x - 2)² + 2 x² + 8x + 16 + 14 = x² - 4x + 4 + 2 x² + 8x + 30 = x² - 4x + 6 x² + 8x - x² + 4x = 6 - 30 12x = - 24 /:12 x = - 2 e) (2x - 3)² = (1 - 2x)² + 2·(4 - 4x) 4x² - 12x + 9 = 1 - 4x + 4x² + 8 - 8x 4x² - 12x + 9 = 4x² - 12x + 9 4x² - 12x - 4x² + 12x = 9 - 9 0 = 0 równanie tożsamościowe x ∈ R f) (4x - 3)(4x + 3) - (x + 1)² = 15·(x - 2)² - 10 - 2x 16x² - 9 - (x² + 2x + 1) = 15·(x² - 4x + 4) - 10 - 2x 16x² - 9 - x² - 2x - 1 = 15x² - 60x + 60 - 10 - 2x 15x² - 2x - 10 = 15x² - 62x + 50 15x² - 2x - 15x² + 62x = 50 + 10 60x = 60 / : 60 x = 1 g) [latex]frac{(1-2x)(1+2x)}{3} + frac{6x^2+2}{5} = frac{3x^2}{10} - frac{(x+5)^2}{2} + frac{11+x^2}{15} / cdot 30 [/latex] [latex]10 cdot (1-2x)(1+2x) + 6 cdot (6x^2+2) = 3 cdot 3x^2 - 15 cdot(x+5)^2 + 2 cdot(11+x^2)}[/latex] [latex]10 cdot (1 - 4x^2) + 36x^2 + 12 = 9x^2 - 15 cdot (x^2+10x + 25) + 22 + 2x^2[/latex] [latex]10 - 40x^2 + 36x^2 + 12 = 9x^2 - 15x^2 - 150x - 375 + 22 + 2x^2[/latex] [latex]-4x^2+22 = -4x^2-150x - 353[/latex] [latex]-4x^2+4x^2+150x = - 353 -22[/latex] [latex]150x = - 375 / : 150[/latex] [latex]x = -frac{375}{150}[/latex] [latex]x = -frac{5}{2}[/latex] [latex]x = - 2,5[/latex] h) x·(x-2) - (x + 1)(x - 3) = 1 + 2x x² - 2x - (x² - 3x + x - 3) = 1 + 2x x² - 2x - (x² - 2x - 3) = 1 + 2x x² - 2x - x² + 2x + 3 = 1 + 2x 3 = 1 + 2x - 2x = 1 - 3 - 2x = - 2 / : (-2) x = 1 Zad. 32 a) x·(x +1) - (x² + 1) < 2·(-x + 3) x² + x - x² - 1 < - 2x + 6 x + 2x < 6 + 1 3x < 7 / : 3 [latex]x < frac{7}{3} [/latex] [latex]x < 2frac{1}{3} [/latex] [latex]x in (-infty; 2frac{1}{3}) [/latex] b) (2x - 3)(2x + 3) - 4 · (4x - 2) > (2x - 5)² 4x² - 9 - 16x² + 8 > 4x² - 20x + 25 -12x² - 1 > 4x² - 20x + 25 -12x² - 1 - 4x² + 20x - 25 > 0 -16x² + 20x - 26 > 0 Δ = 400 - 1664 = -1264 < 0 Δ < 0, czyli funkcja nie ma miejsc zerowych, a = - 16 < 0, czyli ramiona w dół, zatem nierówność nie ma rozwiązań. c) 5 · (x - 1)² > 2 · (x² + 4) + x · (3x + 4) + x 5 · (x² - 2x + 1) > 2x² + 8 + 3x² + 4x + x 5x² - 10x + 5 > 5x² + 5x + 8 5x² - 10x - 5x² - 5x > 8 - 5 - 15x > 3 / : (-15) [latex]x < -frac{3}{15}[/latex] [latex]x < -frac{1}{5}[/latex] [latex]x in (-infty; -frac{1}{5})[/latex] d) [latex]frac{3 cdot (x-1)^2}{2} - frac{2 cdot (x+ 3)^2}{3} < frac{5 cdot (x^2 - 8x - 3)}{6} / cdot 6[/latex] [latex]3 cdot 3 cdot (x - 1)^2 - 2 cdot 2 cdot (x + 3)^2 < 5 cdot (x^2 - 8x - 3) [/latex] [latex]9 cdot (x^2 - 2x + 1) - 4 cdot (x^2+6x + 9) < 5x^2 - 40x - 15 [/latex] [latex]9x^2 - 18x + 9 - 4x^2 - 24x -36 < 5x^2 - 40x - 15 [/latex] [latex]5x^2 - 42x - 27 < 5x^2 - 40x - 15 [/latex] [latex]5x^2 - 42x - 5x^2 +40x < - 15 + 27[/latex] [latex]- 2x < 12 / : (-2)[/latex] [latex]x > -6 [/latex] [latex]x in (-6; +infty) [/latex] e) [latex]frac{(x - 2)^2}{2} - (x + 2)(x- 2) leq 8 - frac{(x+2)^2}{2} / cdot 2[/latex] [latex]x^2 -4x + 4 - 2 cdot (x^2-4) leq 16 - (x^2 +4x +4)[/latex] [latex]x^2 -4x + 4 - 2x^2+8 leq 16 - x^2 -4x -4[/latex] [latex]-x^2-4x+12 leq -x^2-4x+12[/latex] [latex]-x^2-4x+x^2+4x leq 12 - 12[/latex] [latex]0 leq 0[/latex] Nierówność tożsamościowe (bezwarunkowe), czyli x ∈ R Zad. 33 a) [latex]left { {{x + 5 > 4 - 3x} atop {6 cdot (x-1) > 3 + 5x}} ight [/latex] [latex]left { {{x + 3x > 4 - 5} atop {6x-6 > 3 + 5x}} ight [/latex] [latex]left { {{4x > -1 / : 4} atop {6x-5x > 3+ 6}} ight [/latex] [latex]left { {{x > -frac{1}{4}} atop {x > 9}} ight [/latex] Znajdujemy część wspólną wyliczonych przedziałów, zatem: x > 9, czyli x ∈ (9; +∞) b) [latex]left { {{frac{4x}{3}- frac{1}{6} < 2x + 1 / cdot 6} atop {frac{x}{5}-2frac{1}{2} < 2 - frac{x+3}{10} / cdot 10}} ight[/latex] [latex]left { {{8x -1 <12x + 6} atop {2x-25<20 - x - 3}} ight [/latex] [latex]left { {{8x -12x < 6+1} atop {2x-25<- x +17}} ight [/latex] [latex]left { {{-4x < 7 / : (-4)} atop {2x+x<17+25}} ight [/latex] [latex]left { {{x > -frac{7}{4}} atop {3x<42 / : 3}} ight [/latex] [latex]left { {{x > -1frac{3}{4}} atop {x<14}} ight [/latex] [latex]x in ( -1frac{3}{4}; 14)[/latex] c) [latex]left { {{frac{2-5x}{2}+ 3 geq frac{2x+1}{3}- 2x / cdot 6} atop {2 + frac{x+1}{4} leq frac{x-1}{6} + x / cdot 12}} ight[/latex] [latex]left { {{3 cdot (2-5x)+18 geq 2 cdot (2x+1)-12x} atop {24 + 3 cdot (x+1) leq 2 cdot (x -1 ) + 12x}} ight[/latex] [latex]left { {{6-15x+18 geq 4x+2-12x} atop {24 + 3x+3 leq 2x -2+ 12x}} ight[/latex] [latex]left { {{-15x+24 geq -8x+2} atop {3x+27 leq 14x -2}} ight [/latex] [latex]left { {{-15x+8x geq 2-24} atop {3x-14x leq -2-27}} ight [/latex] [latex]left { {{-7x geq -22 / : (-7)} atop {-11x leq -29 / (-11)}} ight [/latex] [latex]left { {{x leq frac{22}{7}} atop {x geq frac{29}{11}}} ight [/latex] [latex]left { {{x leq 3frac{1}{7}} atop {x geq 2frac{7}{11}}} ight [/latex] [latex]x in langle 2frac{7}{11}; 3frac{1}{7} angle[/latex] d) [latex]left { {{(x +1)^2 - 8 > (x -1)^2 +3x-7} atop {(1-3x)^2 - 5x^2 > (2x-1)(2x+1) -10}} ight[/latex] [latex]left { {{x^2+2x+1-8 > x^2-2x+1+3x-7} atop {1-6x+9x^2-5x^2 > 4x^2-1-10}} ight[/latex] [latex]left { {{x^2+2x-7 > x^2+x-6} atop {4x^2-6x+1> 4x^2-11}} ight[/latex] [latex]left { {{x^2+2x-x^2-x > -6+7} atop {4x^2-6x -4x^2> -11-1}} ight[/latex] [latex]left { {{x > 1} atop {-6x > -12 / : (-6)}} ight[/latex] [latex]left { {{x > 1} atop {x < 2}} ight[/latex] [latex]x in (1; 2)[/latex]
