SolveRationalInequalities - 5-masala yechimi | MathModel.uz

2.13-bo'lim : Ratsional tengsizliklar

5. Quyidagi tengsizlikni yeching.

\[u \le \frac{4}{{u - 3}}\]

Yechim

Biz qilishimiz kerak bo'lgan birinchi narsa - tengsizlikning bir tomonida nolni olish va keyin iloji bo'lsa, imkon qadar pay va maxrajni koeffitsientga kiritish.

Shunday qilib, biz birinchi navbatda tengsizlikning bir tomonida nolga erishishimiz kerak.

\[u - \frac{4}{{u - 3}} \le 0\]

Endi biz ikkita atamani bitta ratsional ifodaga birlashtirishimiz kerak.

\[\begin{align*}\frac{{u\left( {u - 3} \right)}}{{u - 3}} - \frac{4}{{u - 3}} & \le 0\\ \frac{{{u^2} - 3u - 4}}{{u - 3}} & \le 0\end{align*}\]

Nihoyat, biz numeratorni faktorga kiritishimiz kerak.

\[\frac{{\left( {u - 4} \right)\left( {u + 1} \right)}}{{u - 3}} \le 0\]

2-qadam

Ushbu bo'lim uchun eslatmalardagi muhokamadan esga olingki, ratsional ifoda faqat surat nol va/yoki maxraj nolga teng bo'lgan ishorani o'zgartirishi mumkin.

Biz surat nol bo'lishini ko'rishimiz mumkin,

\[u = - 1\hspace{0.25in}\hspace{0.25in}u = 4\]

va maxraj nolga teng bo'ladi,

\[u = 3\]

3-qadam

Xuddi ko'phad tengsizliklari bilan qilganimiz kabi, biz qilishimiz kerak bo'lgan narsa, oldingi bosqichdagi nuqtalar orasidagi har bir mintaqadagi test nuqtalarida ratsional ifodani tekshirish. Ratsional ifoda sinov nuqtasidagi belgi bilan bir xil belgiga ega bo'ladi, chunki u faqat shu nuqtalarda belgini o'zgartirishi mumkin.

Mana, oldingi bosqichdagi nuqtalar bilan raqamlar chizig'ining eskizi. Shuningdek, biz raqamlar chizig'ida sinov nuqtasi hisoblarini ham ko'rsatamiz. Mana raqam chizig'i.

$A number line for the inequality with dashed vertical lines at $u = - 1$, $u = 3$ and $u = 4$ dividing it into four regions. The test point $u = - 2$ gives $\frac{{\left( { - 6} \right)\left( { - 1} \right)}}{{ - 5}} < 0$, the test point $u = 0$ gives $\frac{{\left( { - 4} \right)\left( 1 \right)}}{{ - 3}} > 0$, the test point $u = 3.5$, whose computation is boxed and joined to its region by an arrow, gives $\frac{{\left( { - 0.5} \right)\left( {4.5} \right)}}{{0.5}} < 0$, and the test point $u = 5$ gives $\frac{{\left( 1 \right)\left( 6 \right)}}{2} > 0$.$

4-qadam

Biz hozir qilishimiz kerak bo'lgan yagona narsa oldingi bosqichdagi raqamlar qatoridan yechim olishdir. Bu erda javobning tengsizlik va interval belgisi.

\[\require{bbox} \bbox[2pt,border:1px solid black]{\begin{array}{c} u \le - 1\hspace{0.25in}{\mbox{and }}\hspace{0.25in} 3 < u \le 4\\ \left( { - \infty , - 1} \right] \hspace{0.25in} {\mbox{and }}\hspace{0.25in} \left( {3,4} \right]\end{array}}\]

Ushbu muammoning so'nggi nuqtalari bilan ehtiyot bo'ling. Dastlabki tengsizlikda teng belgiga ega bo'lganimiz uchun biz $u = - 1$ va $u = 4$ ni kiritishimiz kerak, chunki u erda surat va shuning uchun ratsional ifoda nolga teng bo'ladi. Biroq, biz $u = 3$ ni qo'sha olmaymiz, chunki u erda maxraj nolga teng va shuning uchun ratsional ifoda o'sha nuqtada nolga bo'linadi!