Absolute value expression

Question

A number, x is more than 12 units from the number 7. Write an expression using absolute value that can be used to determine X's location and solve it

Answer 1

Hi Kt B,

The absolute value of a number is simply the distance a number is from 0 on the number line. So, |4| is 4 because it is 4 units away from 0. Also, |-4| is also 4 because -4 is 4 units from 0.

Your question says "a number, x is more than 12 units from the number 7". This means the distance between x and 7 (distance between also means the difference in the numbers) is more than 12. The inequality that represents this is

|x - 7| > 12

This means that x - 7 > 12 or x - 7 < -12

If x - 7 > 12, then x > 19.

If x - 7 < -12, then x < -5

Hope this helps! :)

x - 7

Answer 2

|x - 7| > 12

We need to solve two cases: (a) when x - 7 is positive, and (b) when x - 7 is negative.

(a)

x - 7 positive ⇒

x - 7 > 12

x > 19

(b)

x - 7 negative ⇒

-(x - 7) > 12

x - 7 < -12 [Note you reverse the inequality when you multiply by -1]

x < -5

The solution is x < -5 or x > 19. Note both solutions put x at more than 12 units from 7.

Answer 3

You can think this through. If x is 12 units from 7 then it is either -5 or -19.

|x+12|= 7

then solve both sides

x+12= 7

x= -5

x+12 = -7

x= -19

SO more than would make the = sign a < than sign

x<-5 and x>-19 Remember since you are changing the sign of the answer in the second answer you have to change the direction of the < sign to. This can be changed to read -19<x<-5