How do you graph the inequality #x^2 – 12x + 32 <12#?
See the explanation
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:

Rewrite the inequality in standard form: (x^2  12x + 32 < 12) (x^2  12x + 20 < 0)

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).

Calculate the discriminant (b^2  4ac) to determine the number of real roots: (b^2  4ac = (12)^2  4(1)(20) = 144  80 = 64)

Since the discriminant is positive, there are two real roots.

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.

Plot the critical points on the number line and shade the appropriate regions to represent where the inequality is true.

The graph of the inequality will be the shaded region on the number line.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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