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Question 0101c
Equations and Inequalities
Inequalities III
Question
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What is the approach to solving exactly the inequality
x
2
+
x
+
4
x
+
2
>
0
?
\frac{x^2 + x + 4}{x + 2} > 0?
x
+
2
x
2
+
x
+
4
>
0
?
Attempt
Select the correct option:
Complete the square for
x
2
+
x
+
4
{x^2 + x + 4}
x
2
+
x
+
4
into the form
(
x
+
p
)
2
+
q
{(x+p)^2 + q}
(
x
+
p
)
2
+
q
to show that it is always positive.
"Cross multiply" by multiplying
x
+
2
{x + 2}
x
+
2
to the RHS of the equation.
Factorize
x
2
+
x
+
4
{x^2 + x + 4}
x
2
+
x
+
4
into the form
(
x
+
p
)
(
x
+
q
)
{(x+p)(x+q)}
(
x
+
p
)
(
x
+
q
)
to find the critical points.
Use the quadratic formula
x
=
−
b
±
b
2
−
4
a
c
2
a
{x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}}
x
=
2
a
−
b
±
b
2
−
4
a
c
for
x
2
+
x
+
4
=
0
{x^2 + x + 4=0}
x
2
+
x
+
4
=
0
to find the critical points.
Answer
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