Answer
$x\le-\dfrac{23}{9} \text{ or } x\ge3$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find all $x$ such that $f(x)
\ge25
,$ with $f(x)=
|2-9x|
,$ solve the inequality
\begin{array}{l}\require{cancel}
|2-9x|\ge25
.\end{array}
Then graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
2-9x\ge25
\\\\\text{OR}\\\\
2-9x\le-25
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
2-9x\ge25
\\\\
-9x\ge25-2
\\\\
-9x\ge23
\\\\\text{OR}\\\\
2-9x\le-25
\\\\
-9x\le-25-2
\\\\
-9x\le-27
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-9x\ge23
\\\\
x\le\dfrac{23}{-9}
\\\\
x\le-\dfrac{23}{9}
\\\\\text{OR}\\\\
-9x\le-27
\\\\
x\ge\dfrac{-27}{-9}
\\\\
x\ge3
.\end{array}
Hence, the solution set is $
x\le-\dfrac{23}{9} \text{ or } x\ge3
.$
