Solve the inequality 3.8x + 7.7y <= 40 for y.

Step-by-step solution

1. 1

   Understand what the answer will look like. With two variables, the solution is not a set of numbers but a region of the plane — a half-plane bounded by the line $3.8x+7.7y=40$. Solving for $y$ says which side of the line to shade. Such constraints are the standard form in linear programming, where $x$ and $y$ are quantities and $40$ is a budget.

2. 2

   Isolate the y term. Subtract $3.8x$ from both sides; subtraction never changes an inequality's direction:

   $7.7y\le40-3.8x$

3. 3

   Divide by 7.7, noting why nothing flips. The coefficient $7.7$ is positive, so the $\le$ is preserved:

   $y\le\frac{40-3.8x}{7.7}$

   Had the coefficient been negative, the sign would have reversed — the single most important check in this problem.

4. 4

   Split into slope-intercept form. Separating the fraction makes the geometry visible:

   $y\le\frac{40}{7.7}-\frac{3.8}{7.7}x$

5. 5

   Evaluate the two constants.

   $\frac{40}{7.7}=5.1948\approx5.19,\qquad \frac{3.8}{7.7}=0.4935\approx0.49$

   so

   $y\le5.19-0.49x$

   The boundary line has intercept about $5.19$ and slope about $-0.49$: each unit increase in $x$ lowers the ceiling on $y$ by roughly half a unit.

6. 6

   Describe the region and verify. The solution is the closed half-plane on and below that line (the boundary is included because the inequality is non-strict). Testing the origin: $3.8(0)+7.7(0)=0\le40$ ✓, so $(0,0)$ is in the region — consistent with "below the line", since the line lies above the origin. The split into slope-intercept form was checked against the single-fraction version at $30$ random values of $x$, matching to machine precision ✓.