1. **Problem statement:** We have an isosceles triangle with two equal sides labeled $x$ and $8$, and the base labeled $10$. We need to find the value of $x$. 2. **Understanding the problem:** In an isosceles triangle, two sides are equal. Here, the two sides are $x$ and $8$, so we set $x = 8$ because the triangle is isosceles. 3. **Check the triangle inequality:** The base is $10$, and the two equal sides are $8$ each. The triangle inequality states that the sum of any two sides must be greater than the third side. 4. **Verification:** $8 + 8 = 16 > 10$, $8 + 10 = 18 > 8$, and $8 + 10 = 18 > 8$. All conditions hold. 5. **Conclusion:** Since the triangle is isosceles with sides $x$ and $8$ equal, the value of $x$ is: $$x = 8$$ This satisfies the triangle properties and the given base length.