Question 12·200 Super-Hard SAT Math Questions·Algebra

Type $y=\frac52-4\left|\frac35x-\frac12\right|$ in Desmos and click its peak. Desmos shows a height of $2.5$. An absolute value cannot be negative, so subtracting four times it makes the left side highest when the bars equal $0$. That highest value is exactly $\frac52$.

For any fixed $k$, the right side is one height. The left side reaches $\frac52$ and every height below it, so the right side must be above $\frac52$ for the graphs never to meet. Write that condition: $\frac98-\frac23k>\frac52$. The inequality is strict because equality gives one solution at the peak.

Use a regression, a Desmos fit for an unknown, to find the boundary. Type $\frac52\sim\frac98-\frac23k_1$. The subscript makes $k_1$ the unknown to fit, and $\sim$ asks Desmos to make the sides equal. Under PARAMETERS, Desmos shows $k_1=-2.0625$.

Type $k_1$ on a new line and click that line’s fraction button. It changes $-2.0625$ to $-\frac{33}{16}$, the exact cutoff. This value makes the right side equal the peak, so it does not give no solutions.

In $\frac98-\frac23k$, making $k$ smaller makes the right side larger because its coefficient is negative. So values below the cutoff put that side above the peak. The equation has no solution exactly when $k<-\frac{33}{16}$. Choice D.