1.4 Solving Absolute Value Equations

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

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

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 +4 +4 𝑥=10 𝑥=−2 4𝑏−5=19 4𝑏−5=−19 +5 +5 +5 +5 4𝑏=24 4𝑏=−14 4𝑏 4 = 24 4 4𝑏 4 = −14 4 𝑏=6 𝑏= −7 2 𝑜𝑟 −3.5

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=3 1− 2 3 𝑣=−3 −1 −1 −1 −1 − 2 3 𝑣=2 − 2 3 𝑣=−4 −3 2 −2 3 𝑣=2 −3 2 −3 2 −2 3 𝑣=−4 −3 2 𝑣=−3 𝑣=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 6 +12 =4 6 2 −2 +12 =4(−2) 24 =24 8 =−8 24=24 8=−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 2𝑥+12=−4𝑥 −2𝑥 −2𝑥 −2𝑥 −2𝑥 12=2𝑥 12=−6𝑥 12 2 = 2𝑥 2 12 −6 = −6𝑥 −6 6=𝑥 −2=𝑥

Ex. 2 When not by Itself 3 3𝑥+9 −10=−4 +10 +10 3 3𝑥+9 =6 +10 +10 3 3𝑥+9 =6 3 3𝑥+9 3 = 6 3 3𝑥+9 =2 3𝑥+9=2 3𝑥+9=−2 −9 −9 −9 −9 3𝑥=−7 3𝑥=−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 = −7 3 3𝑥 3 = −11 3 𝑥= −7 3 𝑜𝑟 −2.3 𝑥= −11 3 𝑜𝑟 −3.7

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

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

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

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

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−6 2𝑏−9=−𝑏+6 −𝑏 −𝑏 +𝑏 +𝑏 𝑏−9=−6 3𝑏−9=6 +9 +9 +9+9 𝑏=3 3𝑏=15 3𝑏 3 = 15 3 𝑏=3 𝑏=5

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+2 2 4𝑤−1 =3 −4𝑤−2 8𝑤−2=12𝑤+6 8𝑤−2=−12𝑤−6 −8𝑤 −8𝑤 +12𝑤 +12𝑤 −2=4𝑤+6 20𝑤−2=−6 −6 −6 +2 +2 −8=4𝑤 20𝑤=−4 −8 4 = 4𝑤 4 20𝑤 20 = −4 20 −2=𝑤 𝑤= −1 5 𝑜𝑟 −.2