Q. 41 - Precalculus Enhanced with Graphing Utilities

Question

Explain why the inequality $x^2+x+1>0$ has all real numbers as the solution set.

Step-by-Step Solution

Answer

• The entire curve lies above the x-axis.
• So, all the values in the domain of the function are the solution of the given inequality.

Step 1. Given Information

The given inequality is $x^2+x+1>0$.

Step 2. Find the Intercepts

• Consider the function $f\left(x\right)=x^2+x+1$.
• The function cannot be factored further.
• There are no x-intercepts of the curve.
• The value of $f\left(0\right)$ is 1.
• So, the y-intercept is 1.
• This implies that the entire curve lies above the x-axis.
• So, all the values in the domain of the function are the solution of the given inequality.