Opener #3 (8/27/10) Solving Equations and Inequalities Solve the following and graph the solution if it is an inequality. 2x + 5 = 17 2. -4x + 7 = 5(x + 2) 3. A = (a + b)h for a 4. 7x – 15 < x + 1 5. -5(x+2) < 3x + 6 6. 4x + 4(2x -1) = 20 7. 5x – 2 < 3 or 2x – 6 < 4 8. – 12 < -3x – 3 and -3x – 3 < 21

Solving Absolute Value Equations & Inequalities

Absolute Value (of x) Symbol lxl The distance x is from 0 on the number line. Always positive Ex: l-3l=3 -4 -3 -2 -1 0 1 2

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

* Plug in answers to check your solutions! Ex: Solve 6x-3 = 15 6x-3 = 15 or 6x-3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

Get the abs. value part by itself first! Ex: Solve 2x + 7 -3 = 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.

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

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”

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

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.

Ex: Solve & graph. Becomes an “and” problem -3 7 8

Solve & graph. Get absolute value by itself first. Becomes an “or” problem -2 3 4

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