Hello!
I'm trying to solve these two questions--
|x-4|<3
and
|5x-2|>5
and I'm not exactly sure how to answer
For question 1 I got this- 1<x<7
and for question 2 I got this-
x>(7)/(5) or x<-(3)/(5)
I'd love some clarification! Thank you so so much!
| x - 4 | < 3
rule -> |x| < a -> x < a, x > -a
x - 4 < 3 and x - 4 > -3
solve x - 4 < 3 -> x < 7
solve x - 4 > - 3 -> x > 1
you have x < 7 and x > 1
combine, 1 < x < 7
#2
remember rule ^^
|5x - 2| > 5
remember the "greater" sign calls for "or" at answer
use the rule for absolute value
5x - 2 > 5 OR 5x - 2 < -5
just solve and you'll get
x > 7/5 , x < -3/5
then using the 'or' you have answer
x < -3/5 or x > 7/5
Thank you so much!
Hello! I'd be happy to help clarify the solutions to these two inequality questions.
For the first question, |x - 4| < 3, we can solve it by considering two cases: when x - 4 is positive and when it is negative.
Case 1: x - 4 ≥ 0 (x ≥ 4)
In this case, the inequality simplifies to x - 4 < 3, which can be solved as follows:
x - 4 < 3
x < 7
So, x is less than 7 in this case.
Case 2: x - 4 < 0 (x < 4)
In this case, the inequality simplifies to -(x - 4) < 3, which can be solved as follows:
-x + 4 < 3
4 - x < 3
4 - 3 < x
1 < x
So, x is greater than 1 in this case.
Combining the solutions from both cases, we find that x is greater than 1 and less than 7, so the final solution is 1 < x < 7.
For the second question, |5x - 2| > 5, we follow the same approach of considering two cases.
Case 1: 5x - 2 > 0 (x > 2/5)
In this case, the inequality simplifies to 5x - 2 > 5, which can be solved as follows:
5x - 2 > 5
5x > 7
x > 7/5
So, x is greater than 7/5 in this case.
Case 2: 5x - 2 < 0 (x < 2/5)
In this case, the inequality simplifies to -(5x - 2) > 5, which can be solved as follows:
-5x + 2 > 5
-5x > 5 - 2
-5x > 3
x < -3/5
So, x is less than -3/5 in this case.
Combining the solutions from both cases, we find that x is either greater than 7/5 or less than -3/5. Therefore, the final solution is x > 7/5 or x < -3/5.
I hope this clears things up for you! Let me know if you have any further questions.