How do you solve and graph #abs(n+2)>=1#?

Answer 1

See a solution process below:

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

#-1 >= n + 2 >= 1#

Subtract #color(red)(2)# from each segment of system of equations to solve for #n# while keeping the system balanced:

#-1 - color(red)(2) >= n + 2 - color(red)(2) >= 1 - color(red)(2)#

#-3 >= n + 0 >= -1#

#-3 >= n >= -1#

Or

#n <= -3#; #n >= -1#

Or, in interval notation:

#(-oo, -3]#; #[-1, +oo)#

To graph this we will draw a vertical lines at #-3# and #-1# on the horizontal axis.

The lines will be a solid lines because the inequality operators contain an "or equal to" clause.

We will shade to the left and right of the lines to show the intervals:

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Answer 2

To solve the inequality ( |n + 2| \geq 1 ), you need to consider two cases:

  1. When ( n + 2 ) is non-negative: In this case, the absolute value of ( n + 2 ) is just ( n + 2 ), so the inequality becomes ( n + 2 \geq 1 ). Solve for ( n ) to get ( n \geq -1 ).

  2. When ( n + 2 ) is negative: In this case, the absolute value of ( n + 2 ) is ( -(n + 2) ), so the inequality becomes ( -(n + 2) \geq 1 ). Solve for ( n ) to get ( n \leq -3 ).

Combining the results from both cases, the solution to the inequality is ( n \leq -3 ) or ( n \geq -1 ).

To graph the solution on a number line, plot an open circle at -1 and shade to the left to represent ( n \geq -1 ), and plot an open circle at -3 and shade to the right to represent ( n \leq -3 ).

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Answer from HIX Tutor

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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