题目

Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use algebra to determine the values of $x$ for which

\begin{align*} \frac{x + 1}{(x - 3)(x + 2)} \leqslant{}& 1 - \frac{2}{x - 3} \end{align*}

(6)

解答

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分母含有 $x-3$ 和 $x+2$ ，不能直接乘过去。先移到一边通分，再用临界点做符号分析。原式中 $x=-2,3$ 不在定义域内。

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The critical values from the denominator are

\begin{align*} x=-2,\qquad x=3 \end{align*}

Now bring all terms to the left-hand side:

\begin{align*} \frac{x+1}{(x-3)(x+2)}-1+\frac{2}{x-3} \leqslant{}&0\\[4mm] \frac{x+1-(x-3)(x+2)+2(x+2)} {(x-3)(x+2)} \leqslant{}&0 \end{align*}

Simplify the numerator:

\begin{align*} x+1-(x-3)(x+2)+2(x+2) =&\, x+1-(x^2-x-6)+2x+4\\[4mm] =&\, -x^2+4x+11 \end{align*}

\begin{align*} \frac{-x^2+4x+11}{(x-3)(x+2)} \leqslant{}&0\\[4mm] \frac{x^2-4x-11}{(x-3)(x+2)} \geqslant{}&0 \end{align*}

Solve the numerator:

\begin{align*} x^2-4x-11=&\,0\\[2mm] x=&\,\frac{4\pm\sqrt{16+44}}{2}\\[2mm] x=&\,2\pm\sqrt{15} \end{align*}

The critical values in order are

\begin{align*} -2,\qquad 2-\sqrt{15},\qquad 3,\qquad 2+\sqrt{15} \end{align*}

Testing the sign of

\begin{align*} \frac{(x-(2+\sqrt{15}))(x-(2-\sqrt{15}))}{(x-3)(x+2)} \end{align*}

gives the non-negative intervals

\begin{align*} x<-2,\qquad 2-\sqrt{15}\leqslant x<3,\qquad x\geqslant 2+\sqrt{15} \end{align*}

Therefore

\begin{align*} \boxed{ x<-2 \quad\text{or}\quad 2-\sqrt{15}\leqslant x<3 \quad\text{or}\quad x\geqslant 2+\sqrt{15} } \end{align*}