How do you graph the inequality #x^2 – 12x + 32 <12#?
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Given Write as: Now solve as a normal quadratic. Bulk a table of values and plot the curve. Remember that
Or if you multiply everything by (-1) you get Notice that this action turned > into < and that also
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To graph the inequality (x^2 - 12x + 32 < 12), follow these steps:
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Rewrite the inequality in standard form: (x^2 - 12x + 32 < 12) (x^2 - 12x + 20 < 0)
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Find the critical points by setting the expression equal to zero and solving for (x): (x^2 - 12x + 20 = 0) This quadratic equation doesn't factor easily, so you can use the quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}) where (a = 1), (b = -12), and (c = 20).
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Calculate the discriminant (b^2 - 4ac) to determine the number of real roots: (b^2 - 4ac = (-12)^2 - 4(1)(20) = 144 - 80 = 64)
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Since the discriminant is positive, there are two real roots.
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Use the critical points to divide the number line into intervals. Test a value from each interval in the original inequality to determine the sign of the expression in that interval.
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Plot the critical points on the number line and shade the appropriate regions to represent where the inequality is true.
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The graph of the inequality will be the shaded region on the number line.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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