We are given that the area of the rectangle is equal to the area of the triangle, and we need to find the value of $x$.

Step 1: Calculate the area of the rectangle

The formula for the area of a rectangle is:

$\text{Area of rectangle} = \text{length} \times \text{width}$

Given:

• Length of the rectangle = $5 + 2x$ inches
• Width of the rectangle = 10 inches

$\text{Area of rectangle} = (5 + 2x) \times 10 = 50 + 20x$

Step 2: Calculate the area of the triangle

The formula for the area of a triangle is:

$\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

Given:

• Base of the triangle = 30 inches
• Height of the triangle = $4x - 10$ inches

$\text{Area of triangle} = \frac{1}{2} \times 30 \times (4x - 10) = 15 \times (4x - 10) = 60x - 150$

Step 3: Set up the equation

Since the areas are equal, we equate the two expressions:

$50 + 20x = 60x - 150$

Step 4: Solve for $x$

First, subtract $20x$ from both sides:

$50 = 40x - 150$

Next, add 150 to both sides:

$200 = 40x$

Now, divide both sides by 40:

$x = \frac{200}{40} = 5$

Step 5: Conclusion

The value of $x$ is $\boxed{5}$.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

1. What happens to the area of the rectangle as $x$ increases?
2. How would you solve if the width of the rectangle was also variable?
3. Can you find the perimeter of the rectangle for $x = 5$?
4. What is the area of the triangle when $x = 5$?
5. How would the equation change if the triangle had a different base?

Tip: Always verify your solution by plugging the value of $x$ back into the original expressions to check for consistency!