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P1 Q1
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11 P1 Q1
2011 H2 Mathematics Paper 1 Question 1
Equations and Inequalities
Answers
−
2
<
x
<
1
{- 2 < x < 1}
−
2
<
x
<
1
Full solutions
x
2
+
x
+
1
x
2
+
x
−
2
<
0
(
x
+
1
2
)
2
+
3
4
(
x
+
2
)
(
x
−
1
)
<
0
\begin{align*} \frac{x^2 + x + 1}{x^2 + x - 2} &< 0 \\ \frac{\left(x + \frac{1}{2}\right)^2 + \frac{3}{4}}{(x + 2)(x - 1)} &< 0 \end{align*}
x
2
+
x
−
2
x
2
+
x
+
1
(
x
+
2
)
(
x
−
1
)
(
x
+
2
1
)
2
+
4
3
<
0
<
0
Since
(
x
+
1
2
)
2
≥
0
∀
x
∈
R
,
{\left(x + \frac{1}{2}\right)^2 \geq 0 \; \forall x \in \mathbb{R}, }
(
x
+
2
1
)
2
≥
0
∀
x
∈
R
,
(
x
+
1
2
)
2
+
3
4
{\left(x + \frac{1}{2}\right)^2 + \frac{3}{4}}
(
x
+
2
1
)
2
+
4
3
is always positive
1
(
x
+
2
)
(
x
−
1
)
<
0
\frac{1}{(x + 2)(x - 1)} < 0
(
x
+
2
)
(
x
−
1
)
1
<
0
−
2
<
x
<
1
■
- 2 < x < 1 \; \blacksquare
−
2
<
x
<
1
■
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