1. ( x+1)/(2x -3) = 2 2x = 3; x = 1,5 , dlatego DR = R { 1,5} x + 1 = 2*(2x -3) x+1 =4x -6 4x - x = 1 +6 3x = 7 x = 7/3 =============== 2. x -2)/(x +1) = (x+1)/(x +2) DR = R {-1; -2} Mamy (x-2)(x+2) =(x+1)² x² - 4 = x² +2x + 1 2x = -5 x = -2,5 ==================== 3. (x² -3x)/ (x² + 1) = 0 ; DR = R, bo x² + 1 > 0 dla dowolnego x ∈ R x² -3x = 0 x(x - 3) = 0 x1 = 0 x2 = 3 ==================== 4. 2/x = 2x ; DR = R {0} 2 = 2x*x 2x² = 2 --> x² = 1 x1 = -1 x2 = 1 ======================= 5. x² + 3x - 1 = 3/x ; DR = R {0} x(x² + 3x - 1) = 3 x³ + 3x² - x -3 = 0 x²(x +3) -( x +3) = 0 ( x +3)*(x² -1) = 0 (x +3)(x-1)(x+1) = 0 x1 = -3 x2 = -1 x3 = 1 =========================== 6. 2x/(x-1) - (2x -3)/(x+1) = 0 DR = R {-1, 1} 2x(x+1) -(x-1)(2x -3) = 0 2x² +2x -(2x²-3x -2x +3) = 0 2x² + 2x - 2x² +5x - 3 = 0 7x = 3 x = 3/7 ============================== 7. (2x - 1)/(x+3) = 4 DR = R {-3} 2x -1 = 4(x +3) 2x -1 = 4x + 12 2x = -13 x = - 6,5 ================================ 8. (x-2)/(x+1) = (x-1)/(x -2) ; DR = R { -1; 2} (x-2)² = (x-1)(x+1) x² -4x + 4 = x² - 1 -4x = -5 x = 5/4 = 1,25 ================= 9. (x² - 5x)/( x² = 4) = 0 ; DR = R x(x -5) = 0 x1 = 0 x2 = 5 ===================== 10. 4/x = x ; DR = R {0} 4 = x² x1 = -2 x2 = 2 ========================= 11. x² + 6x - 1 = 6/x; DR = r { 0} x(x² +6x -1 ) = 6 x³ + 6x² -x - 6 = 0 x²(x +6) -(x + 6) = 0 (x² -1)(x +6) = 0 (x-1)(x+1)(x +6) = 0 x1 = -6 x2 = -1 x3 = 1 =========================== 12. 4/x ≤ x 4/x - x ≤ 0 4/x - x²/x ≤ 0 ( 4 - x²)/x ≤ 0 [(2 -x)(2+x)]/x ≤ 0 x ∈ < -2; 0> u <2; +∞ ) =============================== 13. 1/x > x 1/x - x > 0 1/x - x²/x > 0 (1 -x²)/x > 0 [(1 -x)(1 +x)]/x > 0 x ∈ ( - ∞; -1) u (0; 1) ================================== ==================================
1.x+1/2x-3=2 /*(2x-3) x+1=2(2x-3) x+1=4x-6 x-4x=-6-1 -3x=-7 /:(-3) x=7/3 x=2 1/3 2.x-2/x+1=x+1/x+2 (x-2)*(x+2)=(x+1)*(x+1) x²-4=x²+2x+1 x²-x²-2x=1+4 -2x=5 /:(-2) x=-2 1/2 3.x²-3x/x²+1=0 /*(x²+1) x²-3x=0 x(x-3)=0 x=0 v x=3 4.2/x=2x /*x 2=2x² /:2 x²=1 x²-1=0 (x-1)(x+1)=0 x=1 v x=-1 5.x²+3x-1=3/x /*x x³+3x²-x=3 x³+3x²-x-3=0 x²(x+3)-(x+3)=0 (x+3)(x²-1)=0 (x+3)(x-1)(x+1)=0 6.2x/x-1-2x-3/x+1=0 /*(x²-1) 2x(x+1)-2x-3(x-1)=0 2x²+2x-2x-3x+3=0 2x²-3x+3=0 Δ=9-4*2*3<0 brak rozwiązań 7.2x-1/x+3=4 /*(x+3) 2x-1=4(x+3) 2x-1=4x+12 -2x=13 /:(-2) x=-6 1/2 8.x-2/x+1=x-1/x-2 /*(x+1)(x-2) (x-2)*(x-2)=(x-1)*(x+1) x²-4x+4=x²-1 x²-4x-x²=-1-4 -4x=-5 /:(-4) x=1 1/4 9.x²-5x/x²+4=0 /*(x²+4) x²-5x=0 x(x-5)=0 x=0 v x=5 10.4/x=x /*x 4=x² x²-4=0 (x-2)(x+2)=0 x=2 v x=-2 11.x²+6x-1=6/x /*x x³+6x²-x=6 x³+6x²-x-6=0 x²(x+6)-(x+6)=0 (x+6)(x²-1)=0 (x+6)(x-1)(x+1)=0 x=-6 v x=1 v x=-1 12.4/x≤x /*x 4≤x² x²-4≥0 (x-2)(x+2)≥0 x∈(-∞,-2>u<2,∞) 13.1/x>x /*x 1>x² x²-1<0 (x-1)(x+1)<0 x∈(-∞,-1)u(1,∞)
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