1. Solving Inequalities: Review of Unit 12 Created by: Amanda Hollenbacher 1/30/2005

2. What are inequalities?  Inequalities look just like equations except for 1 small difference:  They contain one of the following symbols Less than or equal to Greater than or equal to Less than Greater than

3. Types of Inequalities  We studied how to solve several different inequalities 1) Basic Inequality Ex. The basic inequality looks like an equation in 1 variable and can be solved in the same manner.

4. Basic Inequality Add 2 to each side: Divide by 3 Because inequalities have a range of solutions, we can represent the solution as a graph. -2 -1 0 1 2 3 4 * The direction of the arrow and the type of circle is determined by the inequality symbol.

5. A Key Idea SymbolCircleArrow ° °

6. Basic Inequality There is one exception to solving inequalities that is different from equations: How is this inequality different from the one we just solved? The above inequality is solved exactly like the previous one, except for one additional step: Note that because we had to divide by a negative value, our inequality symbol was reversed.

7. A Key Idea  For All Inequalities: ANY TIME YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER, THE INEQUALITY SYMBOL GETS REVERSED!

8. Types of Inequalities 2) Absolute Value Inequalities Ex 1. Just like absolute value equations, we have to write 2 different statements to solve an absolute value inequality.

9. Absolute Value Inequalities Write 2 basic inequalities to solve. Note the change of symbol in the 2 nd statement. Solve each inequality. -3 -2 -1 0 1 2 3 4 5 6 Graph the solution!

10. Absolute Value Inequalities Example 2 This example is solved exactly like the previous one. Note the difference in the solution and the graph! -3 -2 -1 0 1 2 3 4 5

11. A Key Idea  When graphing absolute value inequalities: A less than ( ) SYMBOL PRODUCES 2 LINES!

12. Exceptions There are two exceptions to the rules for absolute value inequalities: Because absolute value is always positive, this statement can never be true. There is no solution! Because absolute value is always positive, this statement is true for all values. Any real number is a solution! There is no work to do for these problems. Just remember the rules!!

13. Types of Inequalities 3) Linear Inequalities Ex. 1) The solutions to linear equations can only be represented graphically. The graphs contain the corresponding linear equation, with a shaded region representing all the solutions.

14. Linear Inequalities To solve a linear inequality, you must first graph the corresponding linear equation. -3 -2 -1 1 2 3 3 2 1 -2 Pick a point on either side of the graph. Substitute that point into the inequality. If the statement is true, shade that side of the line. If false, shade the opposite side.

15. Linear Inequalities The only difference between this example and the previous one is how the line is graphed! -3 -2 -1 1 2 3 3 2 1 -2 Note that because the inequality symbol changed, the line is now dotted instead of solid.

16. Practice and Review  Solve and graph each of the following inequalities: 1) 2) 3)