1. 1.4 Solving Absolute Value Equations

2. What We Will Learn Solve absolute value equations
Solve equations with 2 absolute values
Identify special solutions
Story problems

3. Needed Vocab
Absolute value equation: equation that contains an absolute value expression
Extraneous solution: solution that does not satisfy original equation

4. Ex. 1 Solving when Absolute Value by Itself
𝑥−4 =6
4𝑏−5 =19
Steps
1. make a T
2. on left rewrite original with no changes
3. on right rewrite original, but flip the equation sign and make answer negative
4. solve each for variable
5. check for extraneous
𝑥−4=6
𝑥−4=−6
+4 +4
𝑥=10
𝑥=−2
4𝑏−5=19
4𝑏−5=−19
4𝑏=24
4𝑏=−14
4𝑏 4 = 24 4
4𝑏 4 = −14 4
𝑏=6
𝑏= −7 2 𝑜𝑟 −3.5

5. Your Practice 1− 2 3 𝑣 =3 1− 2 3 𝑣=3 1− 2 3 𝑣=−3 −1 −1 −1 −1
1− 2 3 𝑣= − 2 3 𝑣=−3
− − − −1
− 2 3 𝑣= − 2 3 𝑣=−4
− −2 3 𝑣=2 − − −2 3 𝑣=−4 −3 2
𝑣=− 𝑣=6

6. Ex. 5 Special Solutions No solutions:
Make sure absolute value is by itself first
If not, get by itself first
8𝑏+2 =−10
Plug back in
= −2 +12 =4(−2)
24 = =−8
24= =−8
So -2 is extraneous
Extraneous Solutions:
Solve and then plug answers back into original equation and see if works
Could have if a letter on both sides
2𝑥+12 =4𝑥
2𝑥+12=4𝑥 𝑥+12=−4𝑥
−2𝑥 −2𝑥 −2𝑥 −2𝑥
12=2𝑥 =−6𝑥
12 2 = 2𝑥 −6 = −6𝑥 −6
6=𝑥 −2=𝑥

7. Ex. 2 When not by Itself 3 3𝑥+9 −10=−4 +10 +10 3 3𝑥+9 =6
3 3𝑥+9 =6
3 3𝑥+9 3 = 6 3
3𝑥+9 =2
3𝑥+9= 𝑥+9=−2
−9 − −9 −9
3𝑥=− 𝑥=−11
Steps
1. get by itself
Do like a two step equation
Wait until by itself before seeing if no solution
2. all steps from example one
3𝑥 3 = − 𝑥 3 = −11 3
𝑥= −7 3 𝑜𝑟 − 𝑥= −11 3 𝑜𝑟 −3.7

8. Ex. 2 Your Practice −2 5𝑥−1 −3=−11 +3 +3 −2 5𝑥−1 =−8
−2 5𝑥−1 =−8
−2 5𝑥−1 −2 = −8 −2
5𝑥−1 =4
5𝑥−1 =4
5𝑥−1= 𝑥−1=−4
5𝑥= 𝑥=−3
5𝑥 5 = 𝑥 5 = −3 5
𝑥= 𝑥= −3 5 𝑜𝑟 −.6

9. Ex. 3 Story Problems with Max/Min
In a cheerleading competition, the minimum length of a routine is 4 minutes. The maximum length of the routine is 5 minutes. Write an absolute value equation that represents this.
Halfway:
4.5
Distance: .5
𝑥−4.5 =.5
Steps
1. use 𝑥−ℎ𝑎𝑙𝑓𝑤𝑎𝑦 =𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
2. find halfway and plug in
Use 𝑚𝑎𝑥−𝑚𝑖𝑛 2 +𝑚𝑖𝑛
3. find distance and plug in
Use max – halfway

10. Ex. 3 Continued Given the picture, write an absolute value. Halfway
12.5
Distance
2.5
𝑥−12.5 =2.5
Your Practice
For a poetry contest, the minimum length of a poem is 16 lines. The maximum length is 32 lines. Write an absolute value equation.
Halfway: 24
Distance: 8
𝑥−24 =8

11. Exs. 4 on both sides 3𝑥+4 = 𝑥 Steps 1. make a T
2. rewrite original with no changes
3. rewrite original, but flip equal sign and change sign of everything inside SECOND
4. solve for variable
5. check for extraneous since letter on both sides
3𝑥+4=𝑥
3𝑥+4=−𝑥
−3𝑥 −3𝑥
−3𝑥 −3𝑥
4=−2𝑥
4=−4𝑥
4 −2 = −2𝑥 −2
4 −4 = −4𝑥 −4
−2=𝑥
−1=𝑥

12. Your Practice 2𝑏−9 = 𝑏−6 2𝑏−9=𝑏−6 2𝑏−9=−𝑏+6 −𝑏 −𝑏 +𝑏 +𝑏 𝑏−9=−6 3𝑏−9=6
2𝑏−9=𝑏− 𝑏−9=−𝑏+6
−𝑏 −𝑏 𝑏 𝑏
𝑏−9=− 𝑏−9=6
𝑏= 𝑏=15
3𝑏 3 = 15 3
𝑏= 𝑏=5

13. Ex. 6 Number in front of both
Same as example 4, but put absolute value stuff in parenthesis
Do NOT change sign of front numbers
Use distributive prop now
2 4𝑤−1 =3 4𝑤+2
2 4𝑤−1 =3 4𝑤 𝑤−1 =3 −4𝑤−2
8𝑤−2=12𝑤 𝑤−2=−12𝑤−6
−8𝑤 −8𝑤 𝑤 𝑤
−2=4𝑤 𝑤−2=−6
− −
−8=4𝑤 𝑤=−4
−8 4 = 4𝑤 𝑤 20 = −4 20
−2=𝑤 𝑤= −1 5 𝑜𝑟 −.2