Find X in Rectangle with Perimeter 8: Sides (x-3) and (4+2x)

Q: What if one of my side lengths comes out negative?

Great observation! In this problem, when $ x = 1 $, one side is $ x - 3 = -2 $. In pure math, we accept this answer, but in real life, rectangles can't have negative side lengths!

Q: Why do I need to use the distributive property?

The distributive property helps you expand $ 2(1 + 3x) $ into $ 2 + 6x $. This turns a grouped expression into separate terms you can easily solve. It's like unpacking a suitcase!

Q: How do I combine like terms in the parentheses?

Look for terms with the same variable! In $ (4 + 2x) + (x - 3) $, you have constants (4 and -3) and x terms (2x and x). Combine separately: $ 4 - 3 = 1 $ and $ 2x + x = 3x $.

Q: Can I check my answer by plugging it back in?

Absolutely! When $ x = 1 $: Length = $ 4 + 2(1) = 6 $, Width = $ 1 - 3 = -2 $. Perimeter = $ 2(6 + (-2)) = 2(4) = 8 $ ✓

Q: What if I get confused with the signs?

Take it step by step! When you see $ (4 + 2x) + (x - 3) $, remove parentheses carefully: $ 4 + 2x + x - 3 $. Group positives and negatives: $ (4 - 3) + (2x + x) = 1 + 3x $.

Rectangle Perimeter with Algebraic Expressions

The perimeter of the rectangle below is 8.

Calculate x.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's find the value of X!

00:11 In a rectangle, remember that opposite sides are equal.

00:31 The perimeter of a rectangle is the sum of all its sides.

00:39 Now, let's substitute the given values and solve for X.

01:06 Next, we'll group the terms together.

01:29 Now, let's isolate X to find its value.

01:42 And there you have it! That's how we solve this 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 8.

Calculate x.

Step-by-step solution

To find $x$ , we'll use the concept of the rectangle's perimeter.

Step 1: Identify the given expressions for length and width.

Length $L = 4 + 2x$
Width $W = x - 3$
Perimeter $P = 8$

Step 2: Use the formula for the perimeter of a rectangle: $P = 2(L + W)$ .

Substitute the given expressions into the formula:

8 = 2((4 + 2x) + (x - 3))

Step 3: Simplify the equation within the parentheses:

8 = 2(4 + 2x + x - 3)

8 = 2(1 + 3x)

Step 4: Distribute the 2:

8 = 2 \times 1 + 2 \times 3x

8 = 2 + 6x

Step 5: Solve for $x$ :

8 - 2 = 6x

6 = 6x

x = 1

Therefore, the value of $x$ is 1.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Rectangle perimeter equals 2(length + width)
Technique: Substitute expressions: 2((4+2x) + (x-3)) = 8
Check: When x=1, sides are 6 and -2; perimeter is 8 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to double both dimensions
Don't just add length + width = 8! This ignores that rectangles have 4 sides, not 2. The perimeter formula counts each dimension twice because opposite sides are equal. Always use P = 2(L + W).

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

FAQ

Everything you need to know about this question

What if one of my side lengths comes out negative?

Great observation! In this problem, when $x = 1$ , one side is $x - 3 = -2$ . In pure math, we accept this answer, but in real life, rectangles can't have negative side lengths!

Why do I need to use the distributive property?

The distributive property helps you expand $2(1 + 3x)$ into $2 + 6x$ . This turns a grouped expression into separate terms you can easily solve. It's like unpacking a suitcase!

How do I combine like terms in the parentheses?

Look for terms with the same variable! In $(4 + 2x) + (x - 3)$ , you have constants (4 and -3) and x terms (2x and x). Combine separately: $4 - 3 = 1$ and $2x + x = 3x$ .

Can I check my answer by plugging it back in?

Absolutely! When $x = 1$ : Length = $4 + 2(1) = 6$ , Width = $1 - 3 = -2$ . Perimeter = $2(6 + (-2)) = 2(4) = 8$ ✓

What if I get confused with the signs?

Take it step by step! When you see $(4 + 2x) + (x - 3)$ , remove parentheses carefully: $4 + 2x + x - 3$ . Group positives and negatives: $(4 - 3) + (2x + x) = 1 + 3x$ .

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