Two sides of a triangle measure \( 2 x-4 \) and \( 7 x-2 \) units, respectively. Which of these is a possible length for the third side of the triangle? A \( 3 x \) units B \( 6 x \) units \( 5 x+2 \) units D \( 9 x-6 \) units Check Answer

To determine which option represents a possible length for the third side of the triangle, we'll use the **triangle inequality theorem**. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Given: - Side \( a = 2x - 4 \) - Side \( b = 7x - 2 \) - Third side \( c \) is one of the options: \( 3x \), \( 6x \), \( 5x + 2 \), or \( 9x - 6 \) **Step 1: Ensure Positive Lengths** First, ensure that all sides are positive: 1. \( 2x - 4 > 0 \) → \( x > 2 \) 2. \( 7x - 2 > 0 \) → \( x > \frac{2}{7} \) Since \( x > 2 \) is more restrictive, we'll use that. **Step 2: Apply the Triangle Inequality** For sides \( a \), \( b \), and \( c \), the following must hold: 1. \( a + b > c \) 2. \( a + c > b \) 3. \( b + c > a \) Let's determine the range for \( c \): - \( a + b = (2x - 4) + (7x - 2) = 9x - 6 \) - \( |a - b| = |2x - 4 - (7x - 2)| = | -5x - 2 | = 5x + 2 \) Thus, the third side \( c \) must satisfy: \[ 5x + 2 < c < 9x - 6 \] **Step 3: Evaluate Each Option** 1. **Option A: \( 3x \)** - Check \( 5x + 2 < 3x \): \[ 5x + 2 < 3x \Rightarrow 2x < -2 \Rightarrow x < -1 \] But \( x > 2 \), so **invalid**. 2. **Option B: \( 6x \)** - Check \( 5x + 2 < 6x \): \[ 5x + 2 < 6x \Rightarrow x > 2 \] - Check \( 6x < 9x - 6 \): \[ 6x < 9x - 6 \Rightarrow 3x > 6 \Rightarrow x > 2 \] - Both conditions are satisfied for \( x > 2 \). **Valid**. 3. **Option C: \( 5x + 2 \)** - Check \( 5x + 2 < 5x + 2 \): - This is not true as it results in an equality. **Invalid**. 4. **Option D: \( 9x - 6 \)** - Check \( 9x - 6 < 9x - 6 \): - This is not true as it results in an equality. **Invalid**. **Conclusion:** Only **Option B: \( 6x \) units** satisfies the triangle inequality conditions for \( x > 2 \). **Answer:** B) \(6x\) units