cases x2 x 6 0 2×2 3x 20 cases
Question
$\begin{cases} x^2+x-6 \ge 0 \ -2x^2+3x+2>0 \end{cases}$
Basic Answer
Step 1: Solve the first inequality
Factor the quadratic expression:
Set up the inequality:
Determine the critical points by setting each factor to zero:
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.
Combine the intervals:
Step 2: Solve the second inequality
Factor the quadratic expression:
Set up the inequality:
Determine the critical points by setting each factor to zero:
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.
Combine the intervals:
Step 3: Find the intersection of the solutions
The solution to the first inequality is:
The solution to the second inequality is:
The intersection of these intervals is:
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