As given by the question:
There are given that the inequality:
[tex]4x-7\ge\frac{-12x+14}{4}[/tex]Now,
Multiply both sides by 4:
[tex]\begin{gathered} 4x-7\ge\frac{-12x+14}{4} \\ (4x-7)\times4\ge\frac{-12x+14}{4}\times4 \\ (4x-7)\times4\ge-12x+14 \\ 16x-28\ge-12x+14 \end{gathered}[/tex]Then,
Add 28 to both sides in the above equation:
[tex]\begin{gathered} 16x-28\ge-12x+14 \\ 16x-28+28\ge-12x+14+28 \\ 16x\ge-12x+42 \end{gathered}[/tex]Then,
Add 12x to both sides in the above equation:
[tex]\begin{gathered} 16x\ge-12x+42 \\ 16x+12x\ge-12x+42+12x \\ 28x\ge42 \end{gathered}[/tex]Then,
Divide both sides by 28:
[tex]\begin{gathered} 28x\ge42 \\ \frac{28x}{28}\ge\frac{42}{28} \\ x\ge\frac{3}{2} \end{gathered}[/tex]Hence, the value of the given inequality is shown below:
[tex]x\ge\frac{3}{2}[/tex]