How do you solve #abs(2x-1)<9#?

Answer 1

See a solution process below:

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

#-9 < 2x - 1 < 9#
First, add #color(red)(1)# to each segment of the system of inequalities to isolate the #x# term while keeping the system balanced:
#-9 + color(red)(1) < 2x - 1 + color(red)(1) < 9 + color(red)(1)#
#-8 < 2x - 0 < 10#
#-8 < 2x < 10#
Now, divide each side of the equation by #color(red)(2)# to solve for #x# while keeping the equation balanced:
#-8/color(red)(2) < (2x)/color(red)(2) < 10/color(red)(2)#
#-4 < (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) < 5#
#-4 < x < 5#

Or

#x > -4# and #x < 5#

Or, in interval notation:

#(-4, 5)#
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Answer 2

To solve the inequality |2x - 1| < 9, you'll first isolate the absolute value expression and then split the inequality into two cases based on whether the expression inside the absolute value is positive or negative.

Case 1: When 2x - 1 is positive: 2x - 1 < 9

Case 2: When 2x - 1 is negative: -(2x - 1) < 9

Solve each case separately for x, and then combine the solutions.

Case 1: 2x - 1 < 9 2x < 9 + 1 2x < 10 x < 5

Case 2: -(2x - 1) < 9 -2x + 1 < 9 -2x < 9 - 1 -2x < 8 x > -4

Combine the solutions: -4 < x < 5

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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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