Answer
$(-\infty,-\frac{19}{32})$
Work Step by Step
$\frac{1}{3}(x-10)-4x\gt\frac{5}{6}(2x+1)-1$
Multiply both sides of the equation by 6, the least common multiple, to eliminate fractions.
$6(\frac{1}{3}(x-10))-6(4x)\gt6(\frac{5}{6}(2x+1))-6(1)$
Simplify. Apply the distributive property. Combine like terms.
$2(x-10)-24x\gt5(2x+1)-6$
$2x-20-24x\gt10x+5-6$
$-22x-20\gt10x-1$
Subtract 10x from each side. Add 20 to each side.
$-22x-20-10x\gt10x-1-10x$
$-32x-20\gt-1$
$-32x-20+20\gt-1+20$
$-32x\gt19$
Divide both sides by -32. The inequality sign must be reversed when multiplying by a negative.
$-32x\div-32\lt19\div-32$
$x\lt-\frac{19}{32}$
This is written in interval notation as
$(-\infty,-\frac{19}{32})$
Where ) indicates that the solution set for x approaches, but does not include, $-\frac{19}{32})$.