Find the solution set of: 1)x² + 7x+12 > 0 2)m²-7m ≤-10​

Answer: 1) x² + 7x + 12 > 0Step 1: Factor the quadratic expression.x² + 7x + 12 factors to (x + 3)(x + 4).Step 2: Find the roots.The roots are x = -3 and x = -4. These are the points where the quadratic expression equals zero.Step 3: Determine the intervals.The roots divide the number line into three intervals: (-∞, -4), (-4, -3), and (-3, ∞).Step 4: Test each interval.Interval (-∞, -4): Choose a test point, say x = -5. (-5 + 3)(-5 + 4) = (-2)(-1) = 2 > 0. The inequality holds true in this interval.Interval (-4, -3): Choose a test point, say x = -3.5. (-3.5 + 3)(-3.5 + 4) = (-0.5)(0.5) = -0.25 < 0. The inequality is false in this interval.Interval (-3, ∞): Choose a test point, say x = 0. (0 + 3)(0 + 4) = 12 > 0. The inequality holds true in this interval.Step 5: Write the solution set.The solution set is (-∞, -4) ∪ (-3, ∞). This means x is less than -4 or greater than -3.2) m² - 7m ≤ -10Step 1: Move all terms to one side.m² - 7m + 10 ≤ 0Step 2: Factor the quadratic expression.m² - 7m + 10 factors to (m - 2)(m - 5).Step 3: Find the roots.The roots are m = 2 and m = 5.Step 4: Determine the intervals.The roots divide the number line into three intervals: (-∞, 2), (2, 5), and (5, ∞).Step 5: Test each interval.Interval (-∞, 2): Choose a test point, say m = 0. (0 - 2)(0 - 5) = 10 > 0. The inequality is false in this interval.Interval (2, 5): Choose a test point, say m = 3. (3 - 2)(3 - 5) = -2 < 0. The inequality holds true in this interval.Interval (5, ∞): Choose a test point, say m = 6. (6 - 2)(6 - 5) = 4 > 0. The inequality is false in this interval.Step 6: Include the roots.Since the inequality includes "≤", the roots are part of the solution set.Step 7: Write the solution set.The solution set is [2, 5]. This means m is greater than or equal to 2 and less than or equal to 5.Step-by-step explanation:

Answered by NioCharlesUmpar | 2025-07-24