What values of \( x \), on the real number line, satisfy \( -x+2<5 \) and/or \( x+3 \leq 4 \) ?

Let's solve the inequalities step by step. 1. For the first inequality \(-x + 2 < 5\): \[ -x + 2 < 5 \] Subtract 2 from both sides: \[ -x < 3 \] Now, multiply both sides by -1 (remember to flip the inequality sign): \[ x > -3 \] So, the solution for the first inequality is \( x > -3 \). 2. For the second inequality \(x + 3 \leq 4\): \[ x + 3 \leq 4 \] Subtract 3 from both sides: \[ x \leq 1 \] Thus, the solution for the second inequality is \( x \leq 1 \). Now, we need to combine the solutions from both inequalities since we are using "and/or." - The first inequality gives us \( x > -3 \). - The second inequality gives us \( x \leq 1 \). The combined solution includes all \( x \) that are greater than -3 and less than or equal to 1. Therefore, the values of \( x \) that satisfy either inequality are: \[ \boxed{(-3, 1]} \]