2.6 Solving absolute value inequalities

What we will learn Solve absolute value inequalities Story problem

Needed vocab Absolute value inequality: inequality that contains an absolute value expression

Ex. 1 solving when absolute value by itself Solve and graph: 𝑥+7 ≤2 𝑥+7≤2 𝑥+7≥−2 −7 −7 −7 −7 𝑥≤−5 𝑥≥−9 Steps 1. make a T 2. rewrite original with no changes 3. rewrite left, but flip the sign and change sign of answer 4. solve for variable Note: Absolute value has to be greater than or equal to 0. If less than 0, then no solution

Your Practice 10−𝑚 ≥5 10−𝑚≥5 10−𝑚≤−5 −10 −10 −10 −10 −𝑚≥−5 −𝑚≤−15 10−𝑚≥5 10−𝑚≤−5 −10 −10 −10 −10 −𝑚≥−5 −𝑚≤−15 −𝑚 −1 ≥ −5 −1 −𝑚 −1 ≤ −15 −1 𝑚≤5 𝑚≥15 5 15

Ex. 2 when absolute value not by itself 4 2𝑥−5 +1>21 −1 −1 4 2𝑥−5 >20 4 2𝑥−5 4 > 20 4 2𝑥−5 >5 2𝑥−5>5 2𝑥−5<−5 +5 +5 +5 +5 2𝑥>10 2𝑥<0 2𝑥 2 > 10 2 2𝑥 2 < 0 2 𝑥>5 𝑥<0 Steps 1. Get by itself 2. make a T 3. rewrite original with no changes 4. rewrite left, but flip the sign and change sign of answer 5. solve for variable

Your practice 3 𝑑+1 −7≥−1 +7 +7 3 𝑑+1 ≥6 3 𝑑+1 3 ≥ 6 3 𝑑+1 ≥2 +7 +7 3 𝑑+1 ≥6 3 𝑑+1 3 ≥ 6 3 𝑑+1 ≥2 𝑑+1≥2 𝑑+1≤−2 −1 −1 −1 −1 𝑑≥1 𝑑≤−3

Ex. 3 absolute deviation story problem You are buying a new computer. The table shows the prices of computers in a store advertisement. You are willing to pay the mean price with an absolute deviation of at most $100. how many computer prices meet your condition? 𝑥−𝑔𝑖𝑣𝑒𝑛 𝑣𝑎𝑙𝑢𝑒 =𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 Given value is mean Mean = 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑖𝑡𝑒𝑚𝑠 𝑡𝑜𝑡𝑎𝑙 𝑖𝑡𝑒𝑚𝑠 Mean = 6640 10 Mean = 664 𝑥−664 ≤100 𝑥−664≤100 𝑥−664≥−100 +664+664 +664 +664 𝑥≤764 𝑥≥564 six