When dealing with an absolute value, there are two cases that must be analyzed.
Case 1: 4x-9 > 0
Then:
[tex]|4x-9|=4x-9[/tex]Solving the equation for x:
[tex]\begin{gathered} |4x-9|=14 \\ \Rightarrow4x-9=14 \\ \Rightarrow4x=14+9 \\ \Rightarrow4x=23 \\ \Rightarrow x=\frac{23}{4} \end{gathered}[/tex]Case 2: 4x-9 < 0
Then:
[tex]|4x-9|=-(4x-9)[/tex]Soving the equation for x:
[tex]\begin{gathered} |4x-9|=14 \\ \Rightarrow-(4x-9)=14 \\ \Rightarrow-4x+9=14 \\ \Rightarrow-4x=14-9 \\ \Rightarrow-4x=5 \\ \Rightarrow x=-\frac{5}{4} \end{gathered}[/tex]The graph of the function f(x)=|4x-9| is:
Therefore, the solution set is:
[tex]\begin{gathered} x_1=\frac{23}{4} \\ x_2=-\frac{5}{4} \end{gathered}[/tex]
