Q. 39 - Precalculus Enhanced with Graphing Utilities

Question

Show that the inequality ${(x-4)}^2\le 0$ has exactly one 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 on or below the axis will only have the point where the graph is on the horizontal axis, which is only one point.

Step 1. Given information

The given inequality is ${(x-4)}^2\le 0$.

Step 2. Find the intercepts

• Consider the function $f\left(x\right)={(x-4)}^2$.
• $f\left(x\right)=0$ for $x=4$.
• So, the x- intercept is $(4,0)$.
• The value of $f\left(0\right)$ is 16.
• So, the value of y-intercept is $(0,16)$.

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 on or below the axis will only have the point where the graph is on the horizontal axis.