How do you solve #abs(x-3)

Question

How do you solve #abs(x-3)<4#?

Kennedy Adams · Accepted Answer

#-1 < x < 7#

Absolute value functions can be split up into two functions; represented in variable form, it would look like

#| a - b | < c#

becomes

#a - b < c# and #a - b > - c#

So, you have

#| x - 3 | < 4#

can be split up into

#x - 3 < 4# and #x - 3 > - 4#

Now we can solve each inequality to get

#x - 3 < 4# #-># add 3 to both sides

#x - cancel(3) + cancel(3) < 4 + 3#

#x<7#

and

#x - 3 > - 4# #-># add 3 to both sides to get

#x - cancel(3) + cancel(3) > -4 + 3#

#x>-1#

so your answer would be

#x > - 1# #x < 7#

or

#-1 < x < 7#

Eli Assouline · Answer

undefined To solve the inequality |x - 3| < 4, you first isolate the absolute value expression by considering two cases:

Case 1: x - 3 is positive or zero (x - 3 ≥ 0):
In this case, the absolute value expression simplifies to x - 3. So, you have:
x - 3 < 4
Add 3 to both sides:
x < 7

Case 2: x - 3 is negative (x - 3 < 0):
In this case, the absolute value expression becomes -(x - 3), which is the same as -x + 3. So, you have:
-x + 3 < 4
Subtract 3 from both sides:
-x < 1
Multiply both sides by -1 (and reverse the inequality):
x > -1

Therefore, the solution to the inequality |x - 3| < 4 is -1 < x < 7.

How do you solve #abs(x-3)&lt;4#?

How do you solve #abs(x-3)<4#?