Problem 8 - Entrance Test

We need to find the integers x that satisfy both inequalities. First inequality: x^2 - 6x + 8 < 0 Factor the quadratic expression: (x - 2)(x - 4) < 0. This inequality holds when x is between the roots 2 and 4. So, 2 < x < 4. The only integer satisfying this condition is x = 3. Second inequality: |x - 3| <= 1 This absolute value inequality can be rewritten as: -1 <= x - 3 <= 1 Add 3 to all parts of the inequality: -1 + 3 <= x <= 1 + 3 2 <= x <= 4. The integers satisfying this condition are x = 2, 3, 4. Now, we need to find the integers that satisfy *both* conditions. We look for the intersection of the solution sets. Solution set for first inequality: {3} Solution set for second inequality: {2, 3, 4} Intersection: {3} intersection {2, 3, 4} = {3}. There is only 1 integer solution, which is x = 3.