The problem on the image states:

"The width of a rectangle is 12 cm less than the length. The perimeter is 156 cm. Solve for $x$, where $x$ represents the length of the rectangle. Then find the length."

Step-by-step solution:

1. Let $x$ be the length of the rectangle.

   • The width will then be $x - 12$ (since it is 12 cm less than the length).

2. The formula for the perimeter $P$ of a rectangle is: $P = 2 \times (\text{Length} + \text{Width})$ Plug in the values: $156 = 2 \times (x + (x - 12))$

3. Simplify the equation: $156 = 2 \times (2x - 12)$ $156 = 4x - 24$

4. Solve for $x$: $156 + 24 = 4x$ $180 = 4x$ $x = \frac{180}{4} = 45$

So, the length of the rectangle is 45 cm.

5. The width is: $x - 12 = 45 - 12 = 33 \text{ cm}$

Final Answer:

• Length = 45 cm
• Width = 33 cm

Would you like a further breakdown or have any questions on this problem?

Here are 5 related questions to explore:

1. How would the perimeter formula change if the width was 12 cm more than the length?
2. If the perimeter was doubled, how would that affect the length and width?
3. What would the dimensions be if the perimeter was 100 cm?
4. Can you express the width and length relationship using inequalities instead of equations?
5. How does the ratio of width to length affect the area of the rectangle?

Tip: Always double-check if you are solving for the correct variable (e.g., length or width) when given perimeter problems!