Pertidaksamaan Nilai Mutlak, Pecahan, dan Irasional

A. Nilai Mutlak

Untuk  x\in \mathbb{R}  konsep nilai mutlak didefinisikan sebagai

\left | x \right |=\left\{\begin{matrix} x,&jika&x\geq 0\\ \\ \\ -x,&jika&x< 0 \end{matrix}\right..

Untuk  x,\: y\: \in \mathbb{R},

\left | x-y \right |=\left\{\begin{matrix} x-y,&jika&x\geq y\\ \\ \\ -(x-y)=&y-x,&jika&x< y \end{matrix}\right..

Sebagai tambahan

\begin{array}{llll}\\ &&\bullet &Perlu\: diingat\: juga\\\\ &&&1.\quad \left | x.y \right |=\left | x \right |.\left | y \right |\\\\ &&&2.\quad \displaystyle \left | \frac{x}{y} \right |=\frac{\left | x \right |}{\left | y \right |}\\\\ &&&3.\quad \left | x \right |=\sqrt{x^{2}}\end{array}.

\LARGE\fbox{\LARGE\fbox{Contoh Soal}}

1. Dengan menggunakan sifat  \left | x \right |=\sqrt{x^{2}}, buktikan bahwa:

\begin{array}{llll}\\ &&.&\\ &&&a.\quad \left | a.b.c \right |=\left | a \right |.\left | b \right |.\left | c \right |\\\\ &&&b.\quad \displaystyle \left | \frac{a}{b} \right |=\frac{\left | a \right |}{\left | b \right |} \end{array}.

Bukti:

1.a      \left | a.b.c \right |=\sqrt{a^{2}.b^{2}.c^{2}}=\sqrt{a^{2}}.\sqrt{b^{2}}.\sqrt{c^{2}}=\left | a \right |.\left | b \right |.\left | c \right |\qquad \textbf{(terbukti)}.

1.b     \displaystyle \left | \frac{a}{b} \right |=\sqrt{\left ( \frac{a}{b} \right )^{2}}=\frac{\sqrt{a^{2}}}{\sqrt{b^{2}}}=\frac{\left | a \right |}{\left | b \right |}\qquad \textbf{(terbukti)}.

\begin{array}{llll}\\ &&2.&Selesaikanlah\: setiap\: soal\: berikut\: ini!\\ &&&a.\quad 4-x=\left | 7x \right |\\ &&&b.\quad \left | 2x-5 \right |=-7\\ &&&c.\quad 2x+3=\left | 4x+5 \right |\\ &&&d.\quad \left | 5x+3 \right |=\left | 3x+5 \right |\\ &&&e.\quad \left | x-2 \right |=\left | 3-2x \right |\\ &&&f.\quad \displaystyle \left | \frac{x+2}{x-2} \right |=5\\ &&&g.\quad \displaystyle \left | \frac{3x+8}{2x-3} \right |=4 \end{array}.

Moga dapat berlanjut insyaAllah

0 thoughts on “Pertidaksamaan Nilai Mutlak, Pecahan, dan Irasional

Tinggalkan Balasan

Alamat email Anda tidak akan dipublikasikan. Ruas yang wajib ditandai *