1-6 Absolute Value Equations and Inequalities Write and solve absolute value equations and inequalities by applying the definition of absolute value.

Solving an Absolute Value Equation What are the solutions of |x| + 2 = 9 Solve using inverse operations |x| = 7 so… what is x? x = 7 or x = -7 Why?

Key Concept What about |2x – 5| = 13? To solve an equation in the form |A| = b, where A represents a variable expression, solve both A = b and A = -b. When solving, always isolate the absolute value expression first. Do not use inverse operations on what is inside. 2x – 5 =13 and 2x – 5 = -13 x = 9 or x = -4

Practice Solve the following: 2|x + 5| - 2 = 6 |3x – 7| + 3 = 20

Absolute Value Equations Since the absolute value is the distance between a number and zero, an absolute value cannot be negative. Solve 3|2z + 9| + 12 = 10 Subtract 12: 3|2z + 9| = -2 Divide by 3: |2z + 9| = - 2 3 Absolute value cannot be negative so there is no solution Be sure to check for extraneous solutions, meaning a solution which does not satisfy the original equation. Ex: when you solve 3𝑥+2 =4𝑥+5, you get the solutions x = - 3 or x = -1. if you substitute these into the original equation, you will find that x = -3 does not satisfy the equation and x = -1 does. So, x = -3 is extraneous and is not a solution.

Inequalities The same method can be applied to solving absolute value inequalities. 2 2𝑥+4 −3≥9 Start by using inverse operations to isolate the absolute value expression. 2|2𝑥+4|≥12 add 3 |2𝑥+4|≥6 divide by 2 2𝑥+4≥6 𝑜𝑟 2𝑥+4≤−6 Notice that the symbol is reversed when you take the opposite sign. 2𝑥≥2 𝑜𝑟 2𝑥≤−10 subtract 4 𝑥≥1 𝑜𝑟 𝑥≤−5 Graph

Assignment Odds p.46 #19, 21, 43-49, 57-63