1 szkoły ponadpodstawowej

I liceum

I technikum

MATeMAtyka 1. Zakres podstawowy i rozszerzony. Po gimnazjum

a) <svg xmlns="http://www.w3.org/2000/svg" width='0.333em' height='2.400em' viewBox='0 0 333 2400'><path d='M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z'/></svg>​53​x+1<svg xmlns="http://www.w3.org/2000/svg" width='0.333em' height='2.400em' viewBox='0 0 333 2400'><path d='M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z'/></svg>​=2 53​x+1=2  <svg xmlns="http://www.w3.org/2000/svg" width='0.333em' height='2.400em' viewBox='0 0 333 2400'><path d='M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z'/></svg>​−1      lub     53​x+1=−2  <svg xmlns="http://www.w3.org/2000/svg" width='0.333em' height='2.400em' viewBox='0 0 333 2400'><path d='M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z'/></svg>​−1 53​x=1                  lub      53​x=−3 x=35​                    lub      x=−3⋅35​=−5 x∈{−5; 35​}

Rozwiązanie 1

a) ∣5x+2∣&lt;7 5x+2&lt;7    ∣−2      i      5x+2&gt;−7    ∣−2 5x&lt;5   ∣:5             i      5x&gt;−9   ∣:5 x&lt;1                      i      x&gt;−59​ x∈(−59​; 1)

a) ∣2x−4∣+∣3x−6∣=10 ∣2⋅(x−2)∣+∣3⋅(x−2)∣=10 ∣2∣⋅∣x−2∣+∣3∣⋅∣x−2∣=10 2∣x−2∣+3∣x−2∣=10 5∣x−2∣=10   ∣:5 ∣x−2∣=2 x−2=2   ∣+2      lub      x−2=−2   ∣+2 x=4                   lub      x=0 x∈{0; 4}

a) ∣2x−6∣+∣3x−9∣&lt;15 ∣2⋅(x−3)∣+∣3⋅(x−3)∣&lt;15 ∣2∣⋅∣x−3∣+∣3∣⋅∣x−3∣&lt;15 2∣x−3∣+3∣x−3∣&lt;15 5∣x−3∣&lt;15    ∣:5 ∣x−3∣&lt;3 x−3&lt;3   ∣+3      i      x−3&gt;−3   ∣+3 x&lt;6                   i      x&gt;0 x∈(0; 6)

a) ∣6x∣−4x2<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​&gt;0 ∣6x∣−(2x)2<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.88em' viewBox='0 0 400000 1944' preserveAspectRatio='xMinYMin slice'><path d='M983 90
l0 -0
c4,-6.7,10,-10,18,-10 H400000v40
H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7
s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744
c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30
c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722
c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5
c53.7,-170.3,84.5,-266.8,92.5,-289.5z
M1001 80h400000v40h-400000z'/></svg>​&gt;0 ∣6x∣−∣2x∣&gt;0 ∣6∣⋅∣x∣−∣2∣⋅∣x∣&gt;0 6∣x∣−2∣x∣&gt;0 4∣x∣&gt;0   ∣:4 ∣x∣&gt;0 x∈R\{0} (wartość bezwgzlędna dla każdej liczby z wyjątkiem zera jest dodatnia)

a) ∣2x−3∣&lt;5 2x−3&lt;5   ∣+3      i      2x−3&gt;−5  ∣+3 2x&lt;8   ∣:2            i      2x&gt;−2   ∣:2 x&lt;4                     i      x&gt;−1 x∈(−1; 4) liczby całkowite x spełniające tą nieroˊwnosˊcˊ:   0, 1, 2, 3