Points A, B, and C are collinear, with B between A and C. If AB = 3x+2, BC = 5x-1, and AC = 33, what is BC?

Since $B$ is between $A$ and $C$, the segments add up: $AB+BC=AC$. Then $(3x+2)+(5x-1)=33$, so $8x+1=33$, $8x=32$, and $x=4$. The question asks for $BC$, which is $5(4)-1=19$, choice H. Check: $AB=3(4)+2=14$, and $14+19=33$. Choice F ($14$) is the length of $AB$, not $BC$. Choice G ($16$) and choice J ($21$) come from substituting incorrect values of $x$ ($x=3.4$ and $x=4.4$ would be needed). Read carefully which segment the question asks for after solving for the variable.