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$x ^ { 2 } + x + 1 > 0$
$x ^ { 2 } + x + 1 > 0$
Solution of inequality
$x \in \mathbb{R} \left ( \text{It holds for all real numbers} \right )$
$x ^{ 2 } +x+1 > 0$
$x \in \mathbb{R} \left ( \text{It holds for all real numbers} \right )$
 Solve a solution to $x$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } > \color{#FF6800}{ 0 }$
 Convert the inequality to a quadratic equation to find $x_{1}, x_{2}$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$
 Calculate using the quadratic formula 
$x \in \emptyset \left ( \text{Do not have the solution} \right )$
$\color{#FF6800}{ x } \in \emptyset \left ( \text{Do not have the solution} \right )$
 If there is no real root, the left side of the inequality has always a positive value or a negative value depending on coefficient of the leading highest term a(= $1$ ) of $ax^{2}+bx+c=0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } > \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } > \color{#FF6800}{ 0 }$
 Since the coefficient of the leading highest term (= $1$ ) is  (+)  , the left side of the inequality is always  (+)  So this inequality is  TRUE  regardless of the value of $x$
$\color{#FF6800}{ x } \in \mathbb{R} \left ( \text{It holds for all real numbers} \right )$
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Inequality
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