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CIE 9709 2022 March Paper 32 Q1

A Level / CIE / P3

CIE 9709 2022 March Paper 32 Paper · Question 1

题目

Problem

Solve the inequality ∣2x+3∣>3∣x+2∣|2x+3|>3|x+2|.

(4)
题目中文翻译

解不等式 ∣2x+3∣>3∣x+2∣|2x+3|>3|x+2|。

(4)
原卷通用作答要求

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程;仅给出计算器答案而没有过程,不予评分。除非题目另有精度要求,非精确数值答案保留 3 位有效数字,以度为单位的角度保留 1 位小数。

解答

解法一

思路

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两边非负,可以等价平方;整理后把二次式因式分解,取两个根之间的区间。

答题过程

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Squaring both non-negative sides,

(2x+3)2>9(x+2)2.(2x+3)^2>9(x+2)^2.

Expanding and rearranging,

4x2+12x+9> 9x2+36x+36,5x2+24x+27< 0,(5x+9)(x+3)< 0.\begin{align*} 4x^2+12x+9>&\,9x^2+36x+36,\\[4mm] 5x^2+24x+27<&\,0,\\[4mm] (5x+9)(x+3)<&\,0. \end{align*}

The product is negative between its roots. Therefore

−3<x<−95.\boxed{-3<x<-\frac95}.