Question 315·Hard·Algebra

Start with the flexibility guard's boundary, where its two sides tie. Type $3b_1+1\sim8b_1-24$. Here $b_1$ stands for a moisture value, and $\sim$ tells Desmos to find where the sides are equal. Under PARAMETERS, it shows $b_1=5$. This is the cutoff, not yet the allowed range.

At $m=5$, the sides tie. For each increase of $1$ in $m$, $8m-24$ rises by $8$, while $3m+1$ rises by $3$. So values to the right make the right side larger. The inclusive sign $\le$ also allows the tie, giving $m\ge5$.

Now type $-4(2a_1-13)+5\sim3a_1+1$, where $a_1$ stands for a moisture value. This regression finds where the over-drying guard's sides tie. Desmos shows $a_1\approx5.090909$ under PARAMETERS. That's the second cutoff.

As $m$ increases by $1$, $2m-13$ rises by $2$, so multiplying by $-4$ makes the left side fall by $8$. The right side, $3m+1$, rises by $3$. Moving right from the tie makes the left side smaller, as required. Because $<$ is strict, the cutoff itself is excluded.

Type $a_1$ on a new Desmos line and use its fraction button to read the exact cutoff, $\frac{56}{11}$. Since $5=\frac{55}{11}$, this is the farther-right lower limit. The intersection of the guards is all moisture values $m>\frac{56}{11}$. Choice B.