Q. 40 - Precalculus Enhanced with Graphing Utilities

Question

Show that the inequality ${(x-2)}^2>0$ has one real number that is not a solution.

Step-by-Step Solution

Answer

• From the graph, it can be observed that the function lies on or above the horizontal axis.
• So, the solution set of the points above the axis will exclude only the point where the graph is on the horizontal axis.

Step 1. Given Information

The given inequality is ${(x-2)}^2>0$.

Step 2. Find the Intercepts

• Consider the function $f\left(x\right)={(x-2)}^2$
• $f\left(x\right)=0$ for $x=2$.
• So, the x- intercept is $(2,0)$.
• The value of $f\left(0\right)$ is 4.
• So, the value of y-intercept is $(0,4)$.
• The vertex of the function is at $(2,0)$.

Step 3. Plot the Function

• Use the obtained intercepts to plot the function.

• From the graph, it can be observed that the function lies on or above the horizontal axis.
• So, the solution set of the points above the axis will exclude only the point where the graph is on the horizontal axis.