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

2. 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?

3. 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

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

5. 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| =
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.

6. 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|≥ add 3
|2𝑥+4|≥ 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

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