The first inequality is obtained by replacing absolute value brackets with parentheses.$$\left(\frac{1}{4}x-\frac{5}{8}\right)>4$$

Add $\frac{5}{8}$ to both sides.

$$\frac{1}{4}x>4+\frac{5}{8}$$

Simplify the right side of the equation.

$$4+\frac{5}{8}=\frac{32}{8}+\frac{5}{8}=\frac{37}{8}$$

Thus

$$\frac{1}{4}x>\frac{37}{8}$$

Multiply both sides of the inequality by the reciprocal of the coefficient of $x$

$$\left(\frac{4}{1}\right)\left(\frac{1}{4}x\right)>\left(\frac{4}{1}\right)\left(\frac{37}{8}\right)$$

Simplify to get the first solution.

$$x>\frac{37}{2}$$

The second inequality is obtained by replacing the absolute value brackets with parentheses AND putting a negative sign in front of the parentheses.

$$\hspace{0.17em}–\left(\frac{1}{4}x-\frac{5}{8}\right)>4$$

Distribute the negative sign to all terms inside the parentheses.

$$–\hspace{0.17em}\frac{1}{4}x+\frac{5}{8}>4$$

Subtract $\frac{5}{8}$ from both sides.

$$–\hspace{0.17em}\frac{1}{4}\mathrm{x>}4-\frac{5}{8}$$

Simplify the right side of the inequality

$$4-\frac{5}{8}=\frac{32}{8}-\frac{5}{8}=\frac{27}{8}$$

Thus

$$–\hspace{0.17em}\frac{1}{4}x>\frac{27}{8}$$

Multiply both sides by the reciprocal of the coefficient of $x$

IMPORTANT: When multiplying both sides of an inequality by a NEGATIVE number, REVERSE the direction of the inequality symbol.

$$(–\hspace{0.17em}\frac{4}{1})(–\hspace{0.17em}\frac{1}{4}x)<(–\hspace{0.17em}\frac{4}{1})\left(\frac{27}{8}\right)$$

Simplify to get the second solution.

$$x<–\hspace{0.17em}\frac{27}{2}$$