Q. 42 - Precalculus Enhanced with Graphing Utilities

Question

Explain why the inequality $x^2-x+1<0$ has the empty set as solution set.

Step-by-Step Solution

Answer

• The entire curve lies above the x-axis.
• The given inequality requires the values for which the function lies below the horizontal axis.
• So, there is no solution of the 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.
• The given inequality requires the values for which the function lies below the horizontal axis.
• So, there is no solution of the inequality.