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How do you solve #abs(x-3)>=1#?

Answer 1

#2 geq x geq 4#

From #|x-3| geq 1# it follows that #x-3 leq -1# and #x-3 geq 1#

Adding 3 to both sides of the equations we get #x leq 2# and #x geq 4# which are our two solutions.

Also, looking at the graph of #y=|x-3|# we can see which values of y are greater than or equal to #1# and these are indeed the values where x is less than or equal to 2, and where x is greater than or equal to 4.

graph{|x-3| [-10, 10, -5, 5]}

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Answer 2

Julia Adams

To solve the inequality (|x - 3| \geq 1), follow these steps:

Split the inequality into two cases based on the sign inside the absolute value.

Case 1: (x - 3 \geq 1) Case 2: (x - 3 \leq -1)
Solve each case separately for (x).

Case 1: (x - 3 \geq 1) Add 3 to both sides: (x \geq 4)

Case 2: (x - 3 \leq -1) Add 3 to both sides: (x \leq 2)
Combine the solutions from both cases.

The solutions for (x) are: (x \geq 4) or (x \leq 2)

So, the solution to the inequality (|x - 3| \geq 1) is (x \geq 4) or (x \leq 2).

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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.

How do you solve #abs(x-3)&gt;=1#?

How do you solve #abs(x-3)>=1#?