To solve the quadratic inequality \( x^2 + x - 42 \geq 0 \) by factoring, we start by factoring the quadratic expression.
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Factoring the quadratic: We need to find two numbers that multiply to \(-42\) and add up to \(1\) (the coefficient of \(x\)). The numbers that satisfy this are \(7\) and \(-6\).
Thus, we can factor the quadratic as: \[ (x - 6)(x + 7) \geq 0 \]
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Finding the critical points: The critical points occur when the product \((x - 6)(x + 7) = 0\). Solving for \(x\): \[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \] \[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \]
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Analyzing the intervals: We have three intervals to test based on the critical points \(-7\) and \(6\):
- \( (-\infty, -7) \)
- \( (-7, 6) \)
- \( (6, \infty) \)
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Choosing test points:
- For the interval \( (-\infty, -7) \), let's test \( x = -8 \): \[ (-8 - 6)(-8 + 7) = (-14)(-1) = 14 \geq 0 \quad \text{(true)} \]
- For the interval \( (-7, 6) \), let's test \( x = 0 \): \[ (0 - 6)(0 + 7) = (-6)(7) = -42 \geq 0 \quad \text{(false)} \]
- For the interval \( (6, \infty) \), let's test \( x = 7 \): \[ (7 - 6)(7 + 7) = (1)(14) = 14 \geq 0 \quad \text{(true)} \]
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Including critical points: Since the inequality is \(\geq 0\), we include the critical points \(x = -7\) and \(x = 6\).
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Conclusion: The solution set for the inequality is: \[ (-\infty, -7] \cup [6, \infty) \]
Thus, in interval notation, the solutions to the quadratic inequality are: \[ (-\infty, -7] \cup [6, \infty) \]