Math / Geometry

QuestionWrite and solve an inequality to find the values of $x$ for which the perimeter of the rectangle is less than 104.

Studdy Solution

STEP 1

What is this asking? We need to find the values of $x$ that make the perimeter of the rectangle smaller than 104. Watch out! Remember, the perimeter is the total distance around the rectangle, and $x$ must be positive since it represents a length.

STEP 2

1. Define the perimeter formula
2. Set up the inequality
3. Solve the inequality

STEP 3

Alright, let's define the perimeter!
The perimeter of a rectangle is twice the sum of its length and width.
The length is $x + 4$ and the width is $x$ .
So, the perimeter $P$ is $2 \cdot (x + 4 + x)$ .

STEP 4

Let's simplify this formula.
Inside the parentheses, we can combine the $x$ terms: $x + x = 2x$ .
So, the perimeter becomes $P = 2 \cdot (2x + 4)$ .

STEP 5

Now, distribute the 2 to both terms inside the parentheses: $2 \cdot 2x + 2 \cdot 4$ .
This simplifies to $P = 4x + 8$ .
Awesome!

STEP 6

We want the perimeter to be less than 104.
So, we can write the inequality as $4x + 8 < 104$ .
This inequality represents the condition that the perimeter is less than 104.

STEP 7

Let's isolate the $x$ term.
We can start by subtracting 8 from both sides of the inequality: $4x + 8 - 8 < 104 - 8$ .
This simplifies to $4x < 96$ .
We're subtracting 8 from both sides to maintain the balance of the inequality.

STEP 8

Now, let's divide both sides by 4 to get $x$ by itself: $\frac{4x}{4} < \frac{96}{4}$ .
This gives us $x < 24$ .
Dividing by a positive number doesn't change the direction of the inequality!

STEP 9

Since $x$ represents a length, it must be positive.
So, we have $x > 0$ .
Combining this with our previous result, we get $0 < x < 24$ .
This is our final solution for the inequality!

STEP 10

The values of $x$ that make the perimeter of the rectangle less than 104 are $0 < x < 24$ .

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QuestionWrite and solve an inequality to find the values of $x$ for which the perimeter of the rectangle is less than 104.

Studdy Solution

STEP 1

What is this asking? We need to find the values of $x$ that make the perimeter of the rectangle smaller than 104. Watch out! Remember, the perimeter is the total distance around the rectangle, and $x$ must be positive since it represents a length.

STEP 2

1. Define the perimeter formula
2. Set up the inequality
3. Solve the inequality

STEP 3

Alright, let's define the perimeter!
The perimeter of a rectangle is twice the sum of its length and width.
The length is $x + 4$ and the width is $x$ .
So, the perimeter $P$ is $2 \cdot (x + 4 + x)$ .

STEP 4

Let's simplify this formula.
Inside the parentheses, we can combine the $x$ terms: $x + x = 2x$ .
So, the perimeter becomes $P = 2 \cdot (2x + 4)$ .

STEP 5

Now, distribute the 2 to both terms inside the parentheses: $2 \cdot 2x + 2 \cdot 4$ .
This simplifies to $P = 4x + 8$ .
Awesome!

STEP 6

We want the perimeter to be less than 104.
So, we can write the inequality as $4x + 8 < 104$ .
This inequality represents the condition that the perimeter is less than 104.

STEP 7

Let's isolate the $x$ term.
We can start by subtracting 8 from both sides of the inequality: $4x + 8 - 8 < 104 - 8$ .
This simplifies to $4x < 96$ .
We're subtracting 8 from both sides to maintain the balance of the inequality.

STEP 8

Now, let's divide both sides by 4 to get $x$ by itself: $\frac{4x}{4} < \frac{96}{4}$ .
This gives us $x < 24$ .
Dividing by a positive number doesn't change the direction of the inequality!

STEP 9

Since $x$ represents a length, it must be positive.
So, we have $x > 0$ .
Combining this with our previous result, we get $0 < x < 24$ .
This is our final solution for the inequality!

STEP 10

The values of $x$ that make the perimeter of the rectangle less than 104 are $0 < x < 24$ .

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QuestionWrite and solve an inequality to find the values of xxx for which the perimeter of the rectangle is less than 104.

Studdy Solution

STEP 1

What is this asking? We need to find the values of xxx that make the perimeter of the rectangle smaller than 104. Watch out! Remember, the perimeter is the total distance around the rectangle, and xxx must be positive since it represents a length.

STEP 2

1. Define the perimeter formula2. Set up the inequality3. Solve the inequality

STEP 3

Alright, let's **define** the perimeter!The perimeter of a rectangle is *twice* the sum of its length and width.The length is x+4x + 4x+4 and the width is xxx.So, the perimeter PPP is 2⋅(x+4+x)2 \cdot (x + 4 + x)2⋅(x+4+x).

STEP 4

Let's **simplify** this formula.Inside the parentheses, we can combine the xxx terms: x+x=2xx + x = 2xx+x=2x.So, the perimeter becomes P=2⋅(2x+4)P = 2 \cdot (2x + 4)P=2⋅(2x+4).

STEP 5

Now, **distribute** the 2 to both terms inside the parentheses: 2⋅2x+2⋅42 \cdot 2x + 2 \cdot 42⋅2x+2⋅4.This simplifies to P=4x+8P = 4x + 8P=4x+8.Awesome!

STEP 6

We want the perimeter to be *less than* 104.So, we can write the inequality as 4x+8<1044x + 8 < 1044x+8<104.This inequality represents the condition that the perimeter is less than 104.

STEP 7

Let's **isolate** the xxx term.We can start by subtracting 8 from both sides of the inequality: 4x+8−8<104−84x + 8 - 8 < 104 - 84x+8−8<104−8.This simplifies to 4x<964x < 964x<96.We're subtracting 8 from both sides to maintain the balance of the inequality.

STEP 8

Now, let's **divide** both sides by 4 to get xxx by itself: 4x4<964\frac{4x}{4} < \frac{96}{4}44x​<496​.This gives us x<24x < 24x<24.Dividing by a positive number doesn't change the direction of the inequality!

STEP 9

Since xxx represents a length, it *must* be positive.So, we have x>0x > 0x>0.Combining this with our previous result, we get 0<x<240 < x < 240<x<24.This is our **final solution** for the inequality!

STEP 10

The values of xxx that make the perimeter of the rectangle less than 104 are 0<x<240 < x < 240<x<24.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionWrite and solve an inequality to find the values of $x$ for which the perimeter of the rectangle is less than 104.

What is this asking? We need to find the values of $x$ that make the perimeter of the rectangle smaller than 104. Watch out! Remember, the perimeter is the total distance around the rectangle, and $x$ must be positive since it represents a length.

1. Define the perimeter formula
2. Set up the inequality
3. Solve the inequality

Alright, let's define the perimeter!
The perimeter of a rectangle is twice the sum of its length and width.
The length is $x + 4$ and the width is $x$ .
So, the perimeter $P$ is $2 \cdot (x + 4 + x)$ .

Let's simplify this formula.
Inside the parentheses, we can combine the $x$ terms: $x + x = 2x$ .
So, the perimeter becomes $P = 2 \cdot (2x + 4)$ .

Now, distribute the 2 to both terms inside the parentheses: $2 \cdot 2x + 2 \cdot 4$ .
This simplifies to $P = 4x + 8$ .
Awesome!

We want the perimeter to be less than 104.
So, we can write the inequality as $4x + 8 < 104$ .
This inequality represents the condition that the perimeter is less than 104.

Let's isolate the $x$ term.
We can start by subtracting 8 from both sides of the inequality: $4x + 8 - 8 < 104 - 8$ .
This simplifies to $4x < 96$ .
We're subtracting 8 from both sides to maintain the balance of the inequality.

Now, let's divide both sides by 4 to get $x$ by itself: $\frac{4x}{4} < \frac{96}{4}$ .
This gives us $x < 24$ .
Dividing by a positive number doesn't change the direction of the inequality!

Since $x$ represents a length, it must be positive.
So, we have $x > 0$ .
Combining this with our previous result, we get $0 < x < 24$ .
This is our final solution for the inequality!

The values of $x$ that make the perimeter of the rectangle less than 104 are $0 < x < 24$ .