Question

The length of the sides of a triangle are \(x + 1\), \(9 - x\) and \(5x - 3\). The number of values of x for which the triangle is isosceles is:

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

First equate each pair of sides to find candidate x-values, then check the triangle inequality on each candidate since not every algebraic solution gives real side lengths.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept.
A triangle is isosceles when two of its three sides are equal, but before that we always need the three given lengths to actually close into a triangle in the first place. That means every pair of sides must add up to more than the remaining side. We test each of the three possible equal pair equations one at a time and then filter out any case that cannot physically form a triangle.

Step 2: List all pairings and solve each.
Pairing 1: $x + 1 = 9 - x$ gives $2x = 8$, so $x = 4$.
Pairing 2: $x + 1 = 5x - 3$ gives $4x = 4$, so $x = 1$.
Pairing 3: $9 - x = 5x - 3$ gives $6x = 12$, so $x = 2$.
So algebraically there seem to be three candidate values: $1$, $2$ and $4$.

Step 3: Test each candidate against the triangle rule.
For $x=4$: sides are $5, 5, 17$. Smallest two add to $10$, and $10 < 17$, so this figure is impossible. Two short sides of length $5$ simply cannot stretch across a base of $17$.
For $x=1$: sides are $2, 8, 2$. Smallest two add to $4$, and $4 < 8$, so this is impossible too.
For $x=2$: sides are $3, 7, 7$. Check all pairs: $3+7=10>7$ and $7+7=14>3$. Every pair clears the third side, so this triangle is valid.

Step 4: Final Answer.
Only the pairing that gives $x=2$ survives the triangle test, since the other two pairings produce side lengths that cannot form any triangle at all, let alone an isosceles one. So there is only one value of $x$ that makes the triangle isosceles. \[ \boxed{1} \]