Soal-Soal Matematika/Sistem persamaan dan pertidaksamaan

Sistem persamaan

Mutlak

rumus: $|f(x)|=\{\begin{array}{cc}f(x),& \text{jika }f(x)\ge 0,\\ -f(x),& \text{jika }f(x)<0.\end{array}$

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

$$\begin{array}{cc}*|x^2+6x|& =|7x+20|\\ |x^2+6x|& =|7x+20|\\ (x^2+6x{)}^2& =(7x+20{)}^2\\ (x^2+6x{)}^2-(7x+20{)}^2& =0\\ (x^2+6x+7x+20)\cdot (x^2+6x-(7x+20))& =0\\ (x^2+13x+20)\cdot (x^2-x-20)& =0\\ \text{untuk}{x}^2+13x+20& =0\\ x^2+13x+20& =0\\ x& =\frac{-13\pm \sqrt{13^2-4\cdot 1\cdot 20}}{2\cdot 1}\\ x& =\frac{-13\pm \sqrt{169-80}}{2}\\ x& =\frac{-13\pm \sqrt{89}}{2}\\ x_1& =\frac{-13+\sqrt{89}}{2}\text{atau}{x}_2=\frac{-13-\sqrt{89}}{2}\\ \text{untuk}{x}^2-x-20& =0\\ x^2-x-20& =0\\ (x-5)\cdot (x+4)& =0\\ x-5& =0\text{atau}x+4=0\\ x_3& =5& & x_4& =-4\\ *3|x-4|+5|7-x|& =31\\ \text{untuk}|x-4|\text{sebagai berikut:}\\ x-4jikax-4& \ge 0\\ x& \ge 4\\ -(x-4)jikax-4& <0\\ -x+4& & x& <4\\ \text{untuk}|7-x|\text{sebagai berikut:}\\ 7-xjika7-x& \ge 0\\ x& \le 7\\ -(7-x)jika7-x& <0\\ -7+x& & x& >7\\ \text{buatlah grafik daerah arsiran (lihat tabel di bawah ini)}\\ \text{daerah arsiran 1}\\ 3|x-4|+5|7-x|& =31\\ 3(-x+4)+5(7-x)& =31\\ -3x+12+35-5x& =31\\ -8x& =-16\\ x& =2\\ \text{daerah arsiran 2}\\ 3|x-4|+5|7-x|& =31\\ 3(x-4)+5(7-x)& =31\\ 3x-12+35-5x& =31\\ -2x& =8\\ x& =-4\\ \text{daerah arsiran 3}\\ 3|x-4|+5|7-x|& =31\\ 3(x-4)+5(-7+x)& =31\\ 3x-12-35+5x& =31\\ 8x& =78\\ x& =\frac{78}{8}\\ x& =\frac{39}{4}\\ \end{array}$$

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

contoh

selesaikan persamaan akar sebagai berikut:

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

Jawaban

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

Pecahan

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

contoh

selesaikan persamaan pecahan sebagai berikut:

• $\frac{2x-5}{x+14}=\frac{x-4}{x}$

Jawaban

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

Sistem pertidaksamaan

Multak

Akar

Pecahan