1. Opener #3 (8/27/10) Solving Equations and Inequalities
Solve the following and graph the solution if it is an inequality.
2x + 5 = x + 7 = 5(x + 2)
3. A = (a + b)h for a 4. 7x – 15 < x + 1
5. -5(x+2) < 3x x + 4(2x -1) = 20
7. 5x – 2 < 3 or 2x – 6 < 4 8. – 12 < -3x – 3 and -3x – 3 < 21

2. Solving Absolute Value Equations & Inequalities

3. Absolute Value (of x) Symbol lxl
The distance x is from 0 on the number line.
Always positive
Ex: l-3l=3

4. Ex: x = 5
What are the possible values of x?
x = or x = -5

5. To solve an absolute value equation:
Isolate the absolute value function by using opposite operations
- undo addition and subtraction
- undo multiplication and division
2. Determine if there is one, two, or no solutions
- if | | is = to a negative, then there are no solutions
- if | | is = to zero, then one solution
- if | | is = to a positive, then two solutions
3. Write the necessary equations and solve.
- if one solution drop the absolute value and solve
- if two solutions drop the absolute value and write two equations, one equal to the positive value the other to its opposite and solve.

6. * Plug in answers to check your solutions!
Ex: Solve 6x-3 = 15
6x-3 = or 6x-3 = -15
6x = or 6x = -12
x = 3 or x = -2
* Plug in answers to check your solutions!

7. Get the abs. value part by itself first!
Ex: Solve 2x = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
2x = 4 or 2x = -18
x = 2 or x = -9
Check the solutions.

8. Guided Practice 2|x – 4| = 8 2. -3|2x – 5| + 5 = 2

9. Absolute Value Inequalities
Absolute value inequalities are really just compound inequalities
Absolute value inequalities can have no solution, infinite #, or All Real Numbers
Once the Absolute value is isolated you determine if it is an “and” or an “or” using the following rules (Read Ab. Value first)
- Less than is an “and”
- Greater than is an “or”

10. Solving and Graphing Absolute Value Inequalities
Isolate the absolute value expression
- undo addition/subtraction
- undo multiplication/division – if you multiply or divide by a negative flip the sign
2. Determine how many solutions it has, you only have to do work if it’s an infinite number
- if | | is less than 0 or a negative number – No Solution
- if | | is greater than a negative number – ALL REALS
- If less than or greater than a positive number – need to solve

11. Solving Continued…
3. Once you determine it has to be solved – write the compound inequality (and/or) by dropping the absolute value for one and flipping the sign and writing the opposite value on the other side for the second. Example: |ax +b| < c ax + b < c and ax + b > -c 4. Solve each part and graph accordingly.

12. Ex: Solve & graph.
Becomes an “and” problem

13. Solve & graph. Get absolute value by itself first.
Becomes an “or” problem

14. Guided Practice |5x – 4| < 6 2. ½ |x-6| - 2 < 2-5