Soal-Soal Matematika/Sistem persamaan dan pertidaksamaan

Sistem persamaan

Mutlak

rumus: $|f(x)|={\begin{cases}f(x),&{\mbox{jika }}f(x)\geq 0,\\-f(x),&{\mbox{jika }}f(x)<0.\end{cases}}$

sifat

1. |a b| = |a| |b|
2. |a/b| = |a|/|b|, b≠0
3. |x| < b -> -b<x<b
4. |x| ≤ b -> -b≤x≤b
5. |x| > b -> x<-b atau x>b
6. |x| ≥ b -> x≤-b atau x≥b
7. |f(x)| < b -> -b<f(x)<b
8. |f(x)| ≤ b -> -b≤f(x)≤b
9. |f(x)| > b -> f(x)<-b atau f(x)>b
10. |f(x)| ≥ b -> f(x)≤-b atau f(x)≥b
11. |f(x)| < |g(x)| -> (f(x))² < (g(x))²
12. |f(x)| > |g(x)| -> (f(x))² > (g(x))²

contoh

selesaikan persamaan multak sebagai berikut:

• |x²+6x| = |7x+20|
• 3|x-4| + 5|7-x| = 31

Jawaban

$${\begin{aligned}*|x^{2}+6x|&=|7x+20|\\|x^{2}+6x|&=|7x+20|\\(x^{2}+6x)^{2}&=(7x+20)^{2}\\(x^{2}+6x)^{2}-(7x+20)^{2}&=0\\(x^{2}+6x+7x+20)\cdot (x^{2}+6x-(7x+20))&=0\\(x^{2}+13x+20)\cdot (x^{2}-x-20)&=0\\{\text{untuk}}x^{2}+13x+20&=0\\x^{2}+13x+20&=0\\x&={\frac {-13\pm {\sqrt {13^{2}-4\cdot 1\cdot 20}}}{2\cdot 1}}\\x&={\frac {-13\pm {\sqrt {169-80}}}{2}}\\x&={\frac {-13\pm {\sqrt {89}}}{2}}\\x_{1}&={\frac {-13+{\sqrt {89}}}{2}}{\text{atau}}x_{2}={\frac {-13-{\sqrt {89}}}{2}}\\{\text{untuk}}x^{2}-x-20&=0\\x^{2}-x-20&=0\\(x-5)\cdot (x+4)&=0\\x-5&=0{\text{atau}}x+4=0\\x_{3}&=5&&x_{4}&=-4\\*3|x-4|+5|7-x|&=31\\{\text{untuk}}|x-4|{\text{sebagai berikut:}}\\x-4jikax-4&\geq 0\\x&\geq 4\\-(x-4)jikax-4&<0\\-x+4&&x&<4\\{\text{untuk}}|7-x|{\text{sebagai berikut:}}\\7-xjika7-x&\geq 0\\x&\leq 7\\-(7-x)jika7-x&<0\\-7+x&&x&>7\\{\text{buatlah grafik daerah arsiran (lihat tabel di bawah ini)}}\\{\text{daerah arsiran 1}}\\3|x-4|+5|7-x|&=31\\3(-x+4)+5(7-x)&=31\\-3x+12+35-5x&=31\\-8x&=-16\\x&=2\\{\text{daerah arsiran 2}}\\3|x-4|+5|7-x|&=31\\3(x-4)+5(7-x)&=31\\3x-12+35-5x&=31\\-2x&=8\\x&=-4\\{\text{daerah arsiran 3}}\\3|x-4|+5|7-x|&=31\\3(x-4)+5(-7+x)&=31\\3x-12-35+5x&=31\\8x&=78\\x&={\frac {78}{8}}\\x&={\frac {39}{4}}\\\end{aligned}}$$

Daerah Arsiran

x < 4	4 ≤ x < 7	x ≥ 7
-x+4	x-4	x-4
7-x	7-x	-7+x

yang memenuhi sdalah HP = {2, 39/4}

Akar

rumus: $y={\sqrt {f(x)}};f(x)\geq 0$

contoh

selesaikan persamaan akar sebagai berikut:

• ${\sqrt {x^{2}+5}}=7-2x$
• ${\sqrt {x^{2}+7}}=10-2x$

Jawaban

$${\begin{aligned}*{\sqrt {x^{2}+5}}=7-2x\\x^{2}+5&=(7-2x)^{2}\\x^{2}+5&=4x^{2}-28x+49\\3x^{2}-28x+44&=0\\(3x-22)(x-2)&=0\\x&={\frac {22}{3}}{\text{atau}}x=2\\*{\sqrt {x^{2}+7}}&=10-2x\\x^{2}+7&=(10-2x)^{2}\\x^{2}+7&=4x^{2}-40x+100\\3x^{2}-40x+93&=0\\(3x-31)(x-3)&=0\\x&={\frac {31}{3}}{\text{atau}}x=3\\\end{aligned}}$$

Pecahan

rumus: $y={\frac {f(x)}{g(x)}};g(x)\neq 0$

contoh

selesaikan persamaan pecahan sebagai berikut:

• ${\frac {2x-5}{x+14}}={\frac {x-4}{x}}$

Jawaban

$${\begin{aligned}{\frac {2x-5}{x+14}}&={\frac {x-4}{x}}\\x(2x-5)&=(x-4)(x+14)\\2x^{2}-5x&=x^{2}+10x-56\\x^{2}-15x+56&=0\\(x-7)(x-8)&=0\\x&=7{\text{atau}}x=8\\\end{aligned}}$$

Sistem pertidaksamaan

Multak

Akar

Pecahan