(4) \( 1+a-\frac{2 a-1}{2} \) (7) \( \frac{2}{x}+\frac{5}{x+1} \)

Certainly! Let's simplify each of the given expressions step by step. --- ### (4) Simplify \( 1 + a - \frac{2a - 1}{2} \) **Step 1:** Combine like terms by expressing \(1\) and \(a\) with a common denominator. \[ 1 = \frac{2}{2}, \quad a = \frac{2a}{2} \] So, \[ 1 + a = \frac{2}{2} + \frac{2a}{2} = \frac{2 + 2a}{2} \] **Step 2:** Subtract the fraction \(\frac{2a - 1}{2}\): \[ \frac{2 + 2a}{2} - \frac{2a - 1}{2} = \frac{2 + 2a - (2a - 1)}{2} = \frac{2 + 2a - 2a + 1}{2} = \frac{3}{2} \] **Final Simplified Form:** \[ \boxed{\dfrac{3}{2}} \] --- ### (7) Simplify \( \frac{2}{x} + \frac{5}{x + 1} \) **Step 1:** Find a common denominator, which is \(x(x + 1)\). \[ \frac{2}{x} = \frac{2(x + 1)}{x(x + 1)}, \quad \frac{5}{x + 1} = \frac{5x}{x(x + 1)} \] **Step 2:** Combine the fractions: \[ \frac{2(x + 1)}{x(x + 1)} + \frac{5x}{x(x + 1)} = \frac{2x + 2 + 5x}{x(x + 1)} = \frac{7x + 2}{x(x + 1)} \] **Final Simplified Form:** \[ \boxed{\dfrac{7x + 2}{x(x + 1)}} \] --- If you have any further questions or need additional explanations, feel free to ask!