Comparing Rectangle Areas: 30x×(4x+8) vs (7+27x)×5 in Fabric Production

Q: Why can't I just compare the original expressions directly?

You need to expand both expressions first using the distributive property. Only after expanding to $ 120x^2 + 240x $ and $ 35 + 135x $ can you properly analyze which is larger.

Q: How do I know which expression is bigger?

The answer depends on the value of x! For large x values, the quadratic term $ 120x^2 $ dominates. For small x values, the constant 35 and linear term $ 135x $ might make the second expression larger.

Q: Should I always test multiple x values?

Yes! When comparing expressions with variables, test several values including:x = 0 (if allowed)x = 1Small decimals like x = 0.1Larger numbers like x = 10

Q: How do I expand expressions with parentheses?

Use the distributive property: multiply the term outside by each term inside. For $ 30x \times (4x + 8) $, multiply $ 30x \times 4x = 120x^2 $ and $ 30x \times 8 = 240x $.

Variable Dependency with Polynomial Comparisons

In a fabric factory, the possible sizes of fabric are:

$30x\times(4x+8)$

$(7+27x)\times5$

How much more material does the factory need?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Which measure requires more material?

00:03 Let's make sure to open parentheses properly

00:07 The factor will multiply each factor in parentheses

00:10 This is simplifying the first measure

00:13 We'll use the same method to simplify the second measure

00:25 Let's assume X equals 1

00:28 Let's substitute into the measures and see which is larger

00:37 In this case, the first measure

00:41 Now let's assume X equals one-tenth

00:45 Let's substitute into the measures and see which is larger

00:58 This is the first measure

01:05 This is the second measure

01:10 In this case, the second measure requires more material

01:16 Therefore, we cannot determine which measure requires more material

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

In a fabric factory, the possible sizes of fabric are:

$30x\times(4x+8)$

$(7+27x)\times5$

How much more material does the factory need?

Step-by-step solution

We begin by simplifying the two exercises using the distributive property:

We start with the first expression.

$30x\times(4x+8)=$

$30x\times4x+30x\times8=$

$120x^2+240x$

We now address the second expression:

$(7+27x)\times5=$

$5\times7+5\times27x=$

$35+135x$

In order to calculate the expressions, let's assume that in each expression x is equal to 1.

We can now substitute the X value into the equation:

$120x^2+240x=120\times1^2+240\times1=120+240=360$

$35+135\times1=35+135=170$

Hence it seems that the first expression is larger and requires more fabric.

Let's now calculate the expressions assuming that x is less than 1. We substitute this value into each of the expressions as follows: $x=\frac{1}{10}$

$\frac{120}{100}+\frac{240}{10}=1\frac{1}{5}+24=25\frac{1}{5}$

$35+\frac{135}{10}=48.5$

This time the second expression seems to be larger and requires more fabric.

Therefore, it is impossible to determine.

Final Answer

It is not possible to calculate.

Key Points to Remember

Essential concepts to master this topic

Distributive Property: Expand expressions before attempting any numerical comparisons
Substitution Test: Try multiple x values like x=1 gives 360 vs 170
Variable Analysis: Check different values to see which expression dominates ✓

Common Mistakes

Avoid these frequent errors

Assuming one expression is always larger without testing multiple values
Don't just substitute one x value and conclude which is bigger = wrong answer! When x=1, the first expression (360) is larger, but when x=0.1, the second expression (48.5) is larger. Always test multiple values to see if the comparison changes with different x values.

Practice Quiz

Test your knowledge with interactive questions

$$ 140-70= $$

FAQ

Everything you need to know about this question

Why can't I just compare the original expressions directly?

You need to expand both expressions first using the distributive property. Only after expanding to $120x^2 + 240x$ and $35 + 135x$ can you properly analyze which is larger.

How do I know which expression is bigger?

The answer depends on the value of x! For large x values, the quadratic term $120x^2$ dominates. For small x values, the constant 35 and linear term $135x$ might make the second expression larger.

What does it mean that we can't determine the answer?

Since the problem doesn't give us a specific value for x, and different x values give different results, we cannot definitively say which expression represents more fabric. The answer depends on the unknown variable x.

Should I always test multiple x values?

Yes! When comparing expressions with variables, test several values including:

x = 0 (if allowed)
x = 1
Small decimals like x = 0.1
Larger numbers like x = 10

How do I expand expressions with parentheses?

Use the distributive property: multiply the term outside by each term inside. For $30x \times (4x + 8)$ , multiply $30x \times 4x = 120x^2$ and $30x \times 8 = 240x$ .

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