1. **State the problem.** We have a triangle with interior angles labeled $2x-10$, $4x$, and $3x-8$. We need to find $x$. 2. **Use the triangle angle-sum formula.** The interior angles of any triangle add up to $180$ degrees. So we write: $$\left(2x-10\right)+4x+\left(3x-8\right)=180$$ 3. **Combine like terms.** Add the $x$-terms and the constants: $$2x+4x+3x-10-8=180$$ $$9x-18=180$$ 4. **Solve for $x$.** Add $18$ to both sides: $$9x-18+18=180+18$$ $$9x=198$$ Now divide both sides by $9$: $$\frac{9x}{9}=\frac{198}{9}$$ $$\frac{\cancel{9}x}{\cancel{9}}=\frac{\cancel{198}}{\cancel{9}}$$ $$x=22$$ 5. **Check the angle measures.** Substitute $x=22$ back into each angle: $$2x-10=2(22)-10=44-10=34$$ $$4x=4(22)=88$$ $$3x-8=3(22)-8=66-8=58$$ Now add them: $$34+88+58=180$$ This matches the triangle angle sum, so the answer is correct. **Final answer:** $x=22$