Q. 17 - Precalculus Enhanced with Graphing Utilities

Question

In problems 7-22, solve the inequality.      

$x^2-x+1\le 0$

Step-by-Step Solution

Answer

The given inequality has no solution. 

Step 1. Given Information

The given inequality is $x^2-x+1\le 0$.

Step 2. Find the intercepts and vertex

• The value of $f\left(0\right)=1$.
• So, the y-intercept is $(0,1)$.
• There are no real roots of $x^2-x+1=0$, so, there are no x-intercepts of the curve.
• The vertex of the parabola exists at $\frac{-b}{2a}=\frac{-(-1)}{2\left(1\right)}=\frac{1}{2}$.
• Find the value of the function at $x=\frac{1}{2}$.

$\begin{array}{rcl}f\left(\frac{1}{2}\right)& =& {\left(\frac{1}{2}\right)}^2-\left(\frac{1}{2}\right)+1\\ & =& \frac{1}{4}-\frac{1}{2}+1\\ & =& \frac{1-2+4}{4}\\ & =& \frac{3}{4}\end{array}$

• So, the vertex is $(\frac{1}{2},\frac{3}{4})$.

Step 3. Plot the graph

Plot the parabola on a graph using the vertex and intercepts calculated in the previous step.    

Step 4. Find the solution set

• From the graph, there is no region that lies below the x-axis.
• So, the given inequality has no solution.