How do you solve the inequality #abs(x + 2) <=11#?
Determine your boundary conditions to solve for the minimum and maximum values of x, then write out the final inequality
We need to determine the boundaries where this inequality is true. Because we're dealing with an absolute value, the two scenarios look like this:
Notice I flipped the sign for the opposite boundary. This is because we're dealing with the negative solution, and when you flip a sign on an inequality, the direction of the comparator flips as well.
Now, let's solve for x in both expressions:
Now that we know our bounds, we can write it as a single inequality:
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To solve the inequality ( |x + 2| \leq 11 ), you'll consider two cases: when ( x + 2 ) is positive and when it's negative.
Case 1: ( x + 2 \geq 0 ) In this case, the absolute value function is just ( x + 2 ), so the inequality becomes: [ x + 2 \leq 11 ]
Case 2: ( x + 2 < 0 ) In this case, the absolute value function is ( -(x + 2) ), so the inequality becomes: [ -(x + 2) \leq 11 ]
Now, solve each inequality separately.
Case 1: [ x + 2 \leq 11 ] [ x \leq 11 - 2 ] [ x \leq 9 ]
Case 2: [ -(x + 2) \leq 11 ] [ -x - 2 \leq 11 ] [ -x \leq 11 + 2 ] [ -x \leq 13 ] [ x \geq -13 ]
So, combining both cases, the solution to the inequality ( |x + 2| \leq 11 ) is: [ -13 \leq x \leq 9 ]
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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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