Fill in each blank so that the resulting statement is true. The restrictions on the variable in the rational equation below are \( \frac{1}{x-2}-\frac{2}{x+5}=\frac{2 x-1}{x^{2}+3 x-10} \) The restrictions on the variable in the rational equation are \( x \neq \square \) and \( x \neq \square \). (Use ascending order.)

To find the restrictions on the variable \( x \) in the rational equation \[ \frac{1}{x-2}-\frac{2}{x+5}=\frac{2 x-1}{x^{2}+3 x-10}, \] we need to identify the values of \( x \) that would make any denominator equal to zero. 1. **Identify the denominators:** - The first term has a denominator of \( x - 2 \). - The second term has a denominator of \( x + 5 \). - The right side has a denominator of \( x^2 + 3x - 10 \). 2. **Set each denominator to zero and solve for \( x \):** - For \( x - 2 = 0 \): \[ x = 2 \] - For \( x + 5 = 0 \): \[ x = -5 \] - For \( x^2 + 3x - 10 = 0 \), we can factor it: \[ x^2 + 3x - 10 = (x + 5)(x - 2) = 0. \] This gives us the same solutions \( x = 2 \) and \( x = -5 \). 3. **List the restrictions:** The restrictions on the variable in the rational equation are \( x \neq 2 \) and \( x \neq -5 \). Thus, the final answer is: The restrictions on the variable in the rational equation are \( x \neq -5 \) and \( x \neq 2 \).