Solve for x 3(x+2)(x+5) 0 Write the solution using interval notation. Use the union symbol uu to express the solution as a union of disjoint intervals. Finite endpoints of all intervals should be integers. If there are no solutions, use the symbol O/ for the empty set. Use the set notation {a} to represent an isolated solution a.

Question

Solve for  x  3(x+2)(x+5) > 0 Write the solution using interval notation. Use the union symbol  uu to express the solution as a union of disjoint intervals. Finite endpoints of all intervals should be integers. If there are no solutions, use the symbol  O/ for the empty set. Use the set notation  {a} to represent an isolated solution a.

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Set Inequality to Zero: Set the inequality 3(x+2)(x+5) > 0 to zero to find the critical points. This gives us the equations x+2 = 0 and x+5 = 0. Solving for x gives us x = -2 and x = -5.Divide Number Line: Use the critical points to divide the number line into intervals: (-∞, -5), (-5, -2), and (-2, ∞).Test Intervals: Test a number from each interval in the inequality to determine where the inequality is true. For (-∞, -5), choose -6. Substituting -6 into the inequality gives 3(-6+2)(-6+5) = 3(-4)(-1) = 12, which is greater than 0. For (-5, -2), choose -3. Substituting -3 into the inequality gives 3(-3+2)(-3+5) = 3(-1)(2) = -6, which is not greater than 0. For (-2, ∞), choose 0. Substituting 0 into the inequality gives 3(0+2)(0+5) = 3(2)(5) = 30, which is greater than 0.Write Solution: Write the solution set in interval notation. The inequality is true for the intervals (-∞, -5) and (-2, ∞). The solution in interval notation is (-∞, -5) ∪ (-2, ∞).

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