To solve this equation, we first need to solve for the absolute value part:
[tex]\begin{gathered} |4x+7|-5=23 \\ |4x+7|=28 \end{gathered}[/tex]Now, we need to consider two possibilities:
[tex]\begin{gathered} 4x+7\ge0 \\ 4x+7<0 \end{gathered}[/tex]In the first case, the absolute value will return the same as inside the vertical bars:
[tex]\begin{gathered} |4x+7|=28 \\ 4x+7=28 \\ 4x=21 \\ x=\frac{21}{4} \\ x=5.25 \end{gathered}[/tex]In the second case, the absolute value will give the same but with opposite sign:
[tex]\begin{gathered} |4x+7|=28 \\ -(4x+7)=28 \\ 4x+7=-28 \\ 4x=-35 \\ x=-\frac{35}{4} \\ x=-8.75 \end{gathered}[/tex]So, the solutions are:
[tex]\begin{gathered} x=5.25 \\ x=-8.75 \end{gathered}[/tex]