Absolute Value Equations and

InequalitiesOBJECTIVE: TSWBAT write and solve an absolute value inequality

Key Concept

The absolute value of a real number x, written |x|, is the distance from zero on the number line.

|4| = 4|-4| = 4

So

If we have |x| = 5 then what can x be?X can equal 5X can also be -5

Because |5| = 5 and |-5| = 5

a. |x|

b. |x|

c. |x|9

d. |x|

h. |x|

i. |y|= 9

j. |z| = -1

k. |m| = 7

l. |x| = 8

m. |x| = -22

n. |x| = -19

o. |x| = 10

A Little More

|x| = 8x = 8

or x = -8

Tru dat?What if we replaced x with x-2

then |x – 2| = 8So x – 2 = 8 or x – 2 = -8

x = 10 or x = -6

a. Start with |m| = 5

replace m with m-3 |m-3| = 5

m-3 = 5 or m – 3 = -5And m = 8 or m = -2

How about |z + 4| = 12

What is the solution of |2x – 1| = 5?

So we have 2 possibilities for |2x – 1| = 5possibility #1 2x – 1 = 5

2x = 6x = 3

or possibility #22x – 1 = -52x = -4x = -2

So x = 3 or x = -2

Try this

What is the solution of |3x + 2| = 4?again 2 possibilitiespossibility #1 3x + 2 = 4

3x = 2x =

possibility #2 3x + 2 = -43x = -6x = -2

and x = or x = -2

Classwork/Homework 1-6A

1. |x – 4| = 1 2. |2x + 3| = 53. |2x + 7| = 23 4. |3x + 2| = 45. |3x + 2| = -6 6. |5x – 1| = 97. |3x – 5| = -8 8. |x – 3| = 119. |6x – 2| = 10 10. |7x – 10| = 11

Solving multi-Step Absolute Value EquationsWhat is the solution of 3|x + 2| - 1 = 8?

Notice before the absolute value expression was alone. In this problem it is not.Our first job is to get the absolute value expression alone.So we will add 1 then divide by 33|x + 2| - 1 = 8+1 +13|x + 2| = 9|x+2| = 3Now we have 2 possibilities#1 x + 2 = 3 and #2 x + 2 = -3

And x = 1 or x = -5

Your turn use the markers and your desk topWhat is the solution of 2|x + 9| + 3 = 7 ?Subtracting 3 and dividing by 2

2|x + 9| = 4|x + 9| = 2

Now the 2 possililitiesx + 9 = 2 or x + 9 = -2and x = -7 or x = -11

Assignment 1.6 Absolute Value Equations1. |-6x| = 24 2. |2x + 8| - 4 = 123. |x-2| = 4x + 8 4. |x – 3| = 95. |3x| = 18 6. 2|3x – 2|= 147. |2x – 3| = -1 8. |y - 5| - 2 = 109. |2z – 3| = 4z – 1 10. |2y – 4| = 1211. |2x + 5| = 3x + 4 12. -|4 – 8b| = 1213. |3x – 1| + 10 = 25 14. 5|6 – 5x|= 15x – 3515. 2|3x – 7|= 10x - 8