How do you solve and write the following in interval notation: #2 + | x/3 - 1 | > 6#?

Answer 1

#(-\infty,-9)\cup (15,\infty)#

First, subtract two from both sides to get #\abs{\frac{x}{3}-1} >4#
The absolute value can be split into #\frac{x}{3}-1>4# #-(\frac[x][3]-1)>4#
Simplifying for the first equation we get the following #\frac{x}{3}>5# #x>15#
For the second equation #1-\frac{x}{3}>4# #-\frac[x][3]>3# #\frac[x][3]<-3# #x<-9#
So all values less than -9 and greater than 15 satisfy the inequality. This can be written as #(-\infty,-9)\cup (15,\infty)#.

Open brackets are used since infinity is not included as a point and 9 and 15 do not satisfy the inequality.

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Answer 2
To solve the inequality 2 + | x/3 - 1 | > 6 and express the solution in interval notation, we follow these steps: 1. Subtract 2 from both sides: | x/3 - 1 | > 4. 2. Break the absolute value inequality into two separate inequalities: x/3 - 1 > 4 and x/3 - 1 < -4. 3. Solve each inequality separately: - For x/3 - 1 > 4, add 1 to both sides and multiply by 3: x > 15. - For x/3 - 1 < -4, add 1 to both sides and multiply by 3: x < -9. 4. Combine the solutions: x < -9 or x > 15. 5. Write the solution in interval notation: (-∞, -9) ∪ (15, ∞).
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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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