Two sides of a triangle have lengths 4 and 10. If the third side has a length of integer x, how many possible value are there for x?

Answer

To determine how many possible integer values the third side \( x \) of a triangle can take, given the lengths of the other two sides are 4 and 10, we can follow these steps:

### Step 1: Apply the Triangle Inequality Theorem
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

### Step 2: Set Up the Inequalities
For our triangle with sides \( s_1 = 4 \), \( s_2 = 10 \), and \( s_3 = x \), we can set up the following inequalities based on the theorem:

1. \( s_1 + s_2 > s_3 \)
2. \( s_1 + s_3 > s_2 \)
3. \( s_2 + s_3 > s_1 \)

### Step 3: Solve Each Inequality
1. From \( 4 + 10 > x \):
\[ 14 > x \quad \text{or} \quad x < 14 \]

2. From \( 4 + x > 10 \):
\[ x > 10 - 4 \quad \Rightarrow \quad x > 6 \]

3. From \( 10 + x > 4 \):
\[ x > 4 - 10 \quad \Rightarrow \quad x > -6 \quad \text{(This inequality is always true since } x \text{ is positive)} \]

### Step 4: Combine the Results
From the inequalities, we have:
- \( x < 14 \)
- \( x > 6 \)

Thus, we can combine these results to get:
\[ 6 < x < 14 \]

### Step 5: Determine Possible Integer Values
The integer values that satisfy \( 6 < x < 14 \) are:
- 7, 8, 9, 10, 11, 12, 13

### Step 6: Count the Possible Values
The possible integer values for \( x \) are 7, 8, 9, 10, 11, 12, and 13. This gives us a total of 7 possible values.

### Final Answer
The number of possible integer values for \( x \) is **7**.

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