How do you solve |x-4|>3?

2 Answers
Oct 17, 2017

color(blue)(x > 7) or color(green)(x< 1)

Explanation:

Remember this:
if abs(a) < 3
then color(blue)(a < 3) and color(green)(a > -3)

if abs(b) > 3
then color(blue)(b > 3) or color(green)(b <-3)

If that doesn't make sense, draw it out on a number line to convince yourself that it does.

Using that rule we can do this to our inequality:

abs(x-4) > 3
color(blue)(x-4 > 3)
color(green)(x-4 < -3)

color(blue)(x-4 > 3)
color(blue)(x > 7)

color(green)(x-4 < -3)
color(green)(x< 1)

color(blue)(x > 7) or color(green)(x< 1)

Oct 17, 2017

x>7 or x<1

Explanation:

|x-4|>3
Solve the absolute value
We know
either x -4 >3 or x -4<-3

We have two possibilities. The first one is x - 4>3 and the second one is x -4<-3
Solve the first one
x - 4>3
Add color(red)(4) to both sides
x cancel(-4) cancelcolor(red)(+4)>3 color(red)(+4)
x >7

Solve the second possibility
x - 4 <-3
Add color(green)(4) to both sides
x cancel(-4) cancelcolor(green)(+4) < -3 color(green)(+4)
x <1