A Triangle Has Side Lengths Measuring 2 X + 2 Ft , X + 3 Ft , 2x + 2 \, \text{ft}, \, X + 3 \, \text{ft}, 2 X + 2 Ft , X + 3 Ft , And N Ft N \, \text{ft} N Ft .Which Expression Represents The Possible Values Of N N N , In Feet? Express Your Answer In Simplest Terms.A. $x - 1 \

Introduction

In geometry, the triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this problem, we are given a triangle with side lengths measuring $2x + 2 \, \text{ft}, \, x + 3 \, \text{ft},$ and $n \, \text{ft}$. Our goal is to find the possible values of $n$ in feet, expressed in simplest terms.

The Triangle Inequality Theorem

The triangle inequality theorem states that for any triangle with side lengths $a$, $b$, and $c$, the following inequalities must hold:

$$a + b > c$$

$$a + c > b$$

$$b + c > a$$

In our problem, we have side lengths $2x + 2$, $x + 3$, and $n$. We can apply the triangle inequality theorem to these side lengths to find the possible values of $n$.

Applying the Triangle Inequality Theorem

Let's apply the triangle inequality theorem to our side lengths:

1. $2x + 2 + x + 3 > n$
2. $2x + 2 + n > x + 3$
3. $x + 3 + n > 2x + 2$

We can simplify these inequalities to get:

1. $3x + 5 > n$
2. $2x + 2 + n > x + 3$
3. $x + 3 + n > 2x + 2$

Simplifying the Inequalities

Let's simplify the second and third inequalities:

2. $2x + 2 + n > x + 3$

$$2x + n > x + 1$$

$$x + n > 1$$

3. $x + 3 + n > 2x + 2$

$$x + n > 2x - 1$$

$$n > x - 1$$

Finding the Possible Values of n

Now we have three inequalities:

1. $3x + 5 > n$
2. $x + n > 1$
3. $n > x - 1$

We can combine these inequalities to find the possible values of $n$.

Combining the Inequalities

Let's combine the first and second inequalities:

1. $3x + 5 > n$
2. $x + n > 1$

We can rewrite the second inequality as:

$$n > 1 - x$$

Now we have:

1. $3x + 5 > n$
2. $n > 1 - x$

We can combine these inequalities by subtracting the second inequality from the first:

$$3x + 5 - (n - (1 - x)) > 0$$

$$3x + 5 - n + 1 - x > 0$$

$$2x + 6 - n > 0$$

Simplifying the Combined Inequality

Now we have:

$$2x + 6 - n > 0$$

We can rewrite this inequality as:

$$n < 2x + 6$$

Finding the Intersection of the Inequalities

Now we have two inequalities:

1. $n < 2x + 6$
2. $n > x - 1$

We can find the intersection of these inequalities by combining them:

$$x - 1 < n < 2x + 6$$

Conclusion

In this problem, we used the triangle inequality theorem to find the possible values of $n$ in feet. We applied the theorem to the side lengths $2x + 2$, $x + 3$, and $n$ and simplified the resulting inequalities to find the intersection of the possible values of $n$. The final answer is:

$$x - 1 < n < 2x + 6$$

Final Answer

The possible values of $n$ in feet are given by the inequality:

$$x - 1 < n < 2x + 6$$

$$$$

Q&A: Understanding the Triangle Inequality Problem

Q: What is the triangle inequality theorem?

A: The triangle inequality theorem states that for any triangle with side lengths $a$, $b$, and $c$, the following inequalities must hold:

$$a + b > c$$

$$a + c > b$$

$$b + c > a$$

Q: How do we apply the triangle inequality theorem to the given side lengths?

A: We can apply the triangle inequality theorem to the side lengths $2x + 2$, $x + 3$, and $n$ by substituting these values into the inequalities:

1. $2x + 2 + x + 3 > n$
2. $2x + 2 + n > x + 3$
3. $x + 3 + n > 2x + 2$

Q: How do we simplify the inequalities?

A: We can simplify the inequalities by combining like terms and rearranging the expressions:

1. $3x + 5 > n$
2. $2x + 2 + n > x + 3$
3. $x + 3 + n > 2x + 2$

Q: How do we find the possible values of n?

A: We can find the possible values of $n$ by combining the inequalities and solving for $n$.

Q: What is the final answer?

A: The possible values of $n$ in feet are given by the inequality:

$$x - 1 < n < 2x + 6$$

Q: What does this inequality represent?

A: This inequality represents the range of possible values of $n$ in terms of the variable $x$.

Q: How can we use this inequality in real-world applications?

A: This inequality can be used in various real-world applications, such as:

• Designing triangles with specific side lengths
• Calculating the maximum and minimum values of a variable
• Solving optimization problems

Q: What are some common mistakes to avoid when working with the triangle inequality theorem?

A: Some common mistakes to avoid when working with the triangle inequality theorem include:

• Not applying the theorem correctly
• Not simplifying the inequalities properly
• Not considering all possible cases

Q: How can we verify the solution to the triangle inequality problem?

A: We can verify the solution by plugging in different values of $x$ and checking if the resulting values of $n$ satisfy the inequality.

Conclusion

In this article, we have discussed the triangle inequality problem and provided a step-by-step solution to find the possible values of $n$. We have also answered some common questions related to the problem and provided tips for avoiding common mistakes.