#### Answer

The solution is $\Big(-\infty,\dfrac{3}{2}\Big]\cup\Big[\dfrac{7}{2},\infty\Big)$

#### Work Step by Step

$|10-4x|+1\ge5$
Take $1$ to the right side of the inequality:
$|10-4x|\ge5-1$
$|10-4x|\ge4$
Solving this absolute value inequality is equivalent to solving two separate inequalities, which are:
$10-4x\ge4$ and $10-4x\le-4$
$\textbf{Solve the first inequality:}$
$10-4x\ge4$
Take $10$ to the right side:
$-4x\ge4-10$
$-4x\ge-6$
Take $-4$ to divide the right side and reverse the direction of the inequality sign:
$x\le\dfrac{-6}{-4}$
$x\le\dfrac{3}{2}$
$\textbf{Solve the second inequality:}$
$10-4x\le-4$
Take $10$ to the right side:
$-4x\le-4-10$
$-4x\le-14$
Take $-4$ to divide the right side and reverse the direction of the inequality sign:
$x\ge\dfrac{-14}{4}$
$x\ge\dfrac{7}{2}$
Expressing the solution in interval notation:
$\Big(-\infty,\dfrac{3}{2}\Big]\cup\Big[\dfrac{7}{2},\infty\Big)$