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# Module 9

### Absolute Value Equations and Inequalities

Solve for x: |x|=9. x=9 -> |x|=9 - Original Equation. |9|=9 - Let x equal 9. 9=9 - True. Check. |x|=9 - Original Equation. |-9|=9 - Let x equal -9. 9=9 - True.
Solve for x: |6x+3|=9. x=1 or x=-2 -> 6x+3=9 - Subtract 3 from both sides. 6x=6 - Divide 6 from both sides. x=1. 6x+3=-9 - Subtract 3 from both sides. 6x=-12 - divide 6 from both sides. x= -2. Plug in answers to check.
Solve for x: 8|x|+26=21. |x|=-5/8. 8|x|+26=21 - Subtract 26 from both sides. 8|x|=-5 - Divide 8 from both sides. |x| = -5/8. Plug in answer to check.
Solve for x: |(6x+5)/3|=-9. no solution. The absolute value of any expression is never a negative number.
How would you graph |x|<9? By creating a number line with a mark on 9 and -9.
Solve for x: |x|<5. x= {|x|-5<x<5)} or (5,-5). This is the answer because if you were to graph this, you would see that 5 and -5 are 5 units away from 0.
Solve for x: |x-8|<4. Then state which numbers would be plotted on a graph. x= 4<x<12, or (4,12). 4 and 12 would be plotted on a graph. Set |x-8|<4 equal to -4<m-8<4. Then add 8 to all sides. 4<x<12.
Solve for x: |3x+2|+4<15. x=-13/3<x<3 or (-13/3, 3). |3x+2|+4<15 - Subtract 4 from both sides. |3x+2|<11 - Set equal to -11<3x+2<11. Then subtract 2 from both sides. -13<3x<9 - Divide 3 from both sides. -13/3<x<3.
Solve for x: |x-6|>10. x<-4 and x>16. Set equal to x-6<-10 and x-6>10 - add 6 to both sides. x<-4 and x>16.
Can the absolute value of any expression be a negative number? No, it can only be zero or a positive number.