For $-4(2m - 13) + 5 < 3m + 1$, distribute the $-4$, then move all $m$ terms to one side and numbers to the other. For $3m + 1 \le 8m - 24$, do the same. Remember: dividing or multiplying by a positive number does not flip the inequality sign.

Once you have the solution for each inequality, think about the overlap of those ranges on a number line. Which values of $m$ make both inequalities true?

Notice that one inequality uses $<$ and the other uses $\le$. This affects whether the boundary value itself is allowed or not. Keep that in mind when choosing between $>$ and $\ge$ in the answer choices.

In Desmos, use $x$ in place of $m$. Enter the two inequalities on separate lines:

• -4(2x - 13) + 5 < 3x + 1
• 3x + 1 <= 8x - 24

Desmos will shade the regions on the $x$-axis where each inequality is true.

Look along the $x$-axis (this represents $m$). You should see where the shaded regions from both inequalities overlap. The solution set is the part of the $x$-axis where both inequalities are true at the same time. Note the $x$-value where this overlapping region starts.

To find the boundary more precisely, graph the equations y = -4(2x - 13) + 5 and y = 3x + 1, then use Desmos’s intersection feature to find the $x$-coordinate where they are equal. That $x$-value is the boundary from the over-drying guard. Compare it to $5$ to see which requirement is stricter, and then express the final solution as an inequality for $m$.

The moisture content $m$ has to make both inequalities true at the same time:

• Over-drying guard: $-4(2m - 13) + 5 < 3m + 1$
• Flexibility guard: $3m + 1 \le 8m - 24$

The plan:

1. Solve each inequality for $m$.
2. Find the intersection (overlap) of the two solution sets.

We will not pick an answer until we know the overlap.

Start with:

Distribute the $-4$ and combine like terms:

Now move all $m$ terms to one side and constants to the other. Add $8m$ to both sides and subtract $1$ from both sides:

So, to satisfy the over-drying guard alone, $m$ must be large enough that $11m$ is greater than $56$.

Now solve:

Subtract $3m$ from both sides:

Add $24$ to both sides:

Divide both sides by $5$ (a positive number, so the inequality direction stays the same):

This can also be written as $m \ge 5$. So the flexibility guard alone allows any $m$ that is at least $5$.

From the over-drying guard we have:

Divide both sides by $11$ (positive) to solve for $m$:

which is the same as $m > \dfrac{56}{11}$.

From the flexibility guard we have $m \ge 5$.

Now compare the two lower bounds:

• One requires $m$ to be greater than $56/11$.
• The other requires $m$ to be at least $5$.

Since $56/11 \approx 5.09$ is larger than $5$, the stricter requirement is $m > \dfrac{56}{11}$. Any $m$ that satisfies this is automatically $\ge 5$.

So the values of $m$ that meet both conditions are:

which matches answer choice B.