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${cases} x^2+x-6 0 \ -2x^2+3x+2>0 {cases}$See answer

cases x2 x 6 0 2×2 3x 20 cases

Question

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$\begin{cases} x^2+x-6 \ge 0 \ -2x^2+3x+2>0 \end{cases}$

Basic Answer

Step 1: Solve the first inequality

  1. Factor the quadratic expression:

  2. Set up the inequality:

  3. Determine the critical points by setting each factor to zero:

  4. Use a sign chart to determine where the product is non-negative:

    • For , both factors are negative, so the product is positive.
    • For , one factor is negative and the other is positive, so the product is negative.
    • For , both factors are positive, so the product is positive.
  5. Combine the intervals:

Step 2: Solve the second inequality

  1. Factor the quadratic expression:

  2. Set up the inequality:

  3. Determine the critical points by setting each factor to zero:

  4. Use a sign chart to determine where the product is positive:

    • For , both factors are negative, so the product is negative.
    • For , one factor is negative and the other is positive, so the product is positive.
    • For , both factors are positive, so the product is negative.
  5. Combine the intervals:

Step 3: Find the intersection of the solutions

  1. The solution to the first inequality is:

  2. The solution to the second inequality is:

  3. The intersection of these intervals is:

Final Answer

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