Rectangle Perimeter Problem: Solve for X When Sides are (x-10) and (4x-8)

Q: Why do I need to distribute 2(x-10) instead of just writing 2x-10?

The distributive property requires you to multiply every term inside the parentheses. So 2(x-10) = 2x - 20, not 2x - 10. Missing this step will give you the wrong answer!

Q: Can I solve this by setting up the equation as 2x + 2(4x-8) - 2(10) = 24?

Be careful with your algebra! It's cleaner to write $ 2(x-10) + 2(4x-8) = 24 $ and then distribute. Your approach could work but risks sign errors.

Q: What if my final answer doesn't make the rectangle realistic?

In pure math problems, focus on solving the equation correctly. Even if x = 6 gives a side length of -4, that's the mathematically correct answer to the given equation.

Perimeter Equations with Variable Expressions

The perimeter of the rectangle below is 24.

Calculate x.

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 The perimeter of the rectangle equals the sum of its sides

00:08 Opposite sides are equal in a rectangle, therefore each side appears twice

00:11 We'll substitute the perimeter value according to the given data, and solve for X

00:14 Group terms

00:17 Isolate X

00:19 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The perimeter of the rectangle below is 24.

Calculate x.

Step-by-step solution

To solve this problem, we'll use the perimeter of a rectangle formula:

Step 1: Identify the side lengths.
Step 2: Use the perimeter formula to set up the equation.
Step 3: Solve for $x$ .

Now, let's work through each step:

Step 1: The side lengths are $x - 10$ and $4x - 8$ .
Step 2: The perimeter $P$ is given by $2(x - 10) + 2(4x - 8) = 24$ .

Simplify the equation:

2(x - 10) + 2(4x - 8) = 24

2x - 20 + 8x - 16 = 24

Combine like terms:

10x - 36 = 24

Step 3: Solve for $x$ :

10x = 24 + 36

10x = 60

x = 6

Therefore, the value of $x$ is $\boxed{6}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Rectangle perimeter equals 2(length + width) or 2l + 2w
Technique: Distribute first: 2(x-10) becomes 2x - 20
Check: Substitute x = 6: sides are -4 and 16, perimeter = 2(-4) + 2(16) = 24 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to double both side lengths
Don't just add (x-10) + (4x-8) = 24, giving 5x - 18 = 24 and x = 8.4! This ignores that rectangles have TWO of each side length. Always use P = 2(side1) + 2(side2) or P = 2(side1 + side2).

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side DC has a length of 1.5 cm and side AD has a length of 9.5 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

What if I get a negative side length when I substitute my answer?

Check your work! In this problem, when x = 6, one side is (6-10) = -4. A negative side length doesn't make physical sense, but mathematically the equation is still correct since the perimeter formula accounts for this.

Why do I need to distribute 2(x-10) instead of just writing 2x-10?

The distributive property requires you to multiply every term inside the parentheses. So 2(x-10) = 2x - 20, not 2x - 10. Missing this step will give you the wrong answer!

Can I solve this by setting up the equation as 2x + 2(4x-8) - 2(10) = 24?

Be careful with your algebra! It's cleaner to write $2(x-10) + 2(4x-8) = 24$ and then distribute. Your approach could work but risks sign errors.

How do I know which expression goes with which side?

Look at the diagram carefully! The top and bottom sides are labeled $x-10$ , while the left and right sides are labeled $4x-8$ . Opposite sides of rectangles are equal.

What if my final answer doesn't make the rectangle realistic?

In pure math problems, focus on solving the equation correctly. Even if x = 6 gives a side length of -4, that's the mathematically correct answer to the given equation.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Rectangles questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime