题目

Problem

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

Solutions relying on calculator technology are not acceptable.

Solve the inequality

\begin{align*} 4x^2-3x+7\ge 4x+9. \end{align*}

(4)

解答

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先把所有项移到同一边，得到二次不等式。二次项系数为正，所以图像开口向上，解集在两个根的外侧。

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\begin{align*} 4x^2-3x+7&\ge 4x+9\\ 4x^2-7x-2&\ge 0. \end{align*}

Factorise:

\begin{align*} 4x^2-7x-2 =&\,(4x+1)(x-2). \end{align*}

So the critical values are

\begin{align*} x=-\frac14 \quad\text{and}\quad x=2. \end{align*}

Since the quadratic opens upwards,

\begin{align*} 4x^2-7x-2\ge0 \end{align*}

outside the two roots. Therefore

\begin{align*} x\le -\frac14 \quad\text{or}\quad x\ge2. \end{align*}