To find the value of $x$ that makes the areas of the two rectangles equal, we can set up an equation by calculating the areas of both rectangles and then equating them.

Step 1: Calculate the area of the first rectangle

• Height = 3
• Width = $x + 2$

So, the area of the first rectangle is: $\text{Area}_1 = 3 \times (x + 2) = 3(x + 2)$

Step 2: Calculate the area of the second rectangle

• Height = $2x + 2$
• Width = 2

So, the area of the second rectangle is: $\text{Area}_2 = 2 \times (2x + 2) = 2(2x + 2)$

Step 3: Set the areas equal to each other

$3(x + 2) = 2(2x + 2)$

Step 4: Solve for $x$

Expand both sides of the equation: $3x + 6 = 4x + 4$

Move all terms involving $x$ to one side and constants to the other: $3x - 4x = 4 - 6$ $-x = -2$

Solve for $x$: $x = 2$

So, the value of $x$ that makes the areas of the two rectangles equal is $x = 2$.

Would you like more details on the solution or clarification on any steps?

Here are five related questions:

1. How do you solve for $x$ if the height or width is given as a quadratic expression?
2. What is the formula for the area of a rectangle?
3. How would you approach this problem if given different geometric shapes?
4. Can you solve similar problems involving squares or triangles?
5. How can we verify the result by calculating the areas for $x = 2$?

Tip: Always expand expressions carefully to avoid errors when solving equations.