2. Equations with Inequalities Solving Equations with Inequalities is just like solving any other equations that have variables. The main difference is that a regular equation with a variable has only one answer, an equation with an inequality has a range of answers. Regular Equation x - 4 = 3 + 4 + 4 x = 7 The answer is one distinct number Inequality Equation x - 4 > 3 + 4 + 4 x > 7 The answer is the range of numbers greater than seven Let’s graph this answer on a number line. 5 7 9 11 13 15 17 Notice the bold section of the line and that the circle on the seven is open.

3. More Equations with Inequalities x + 8 2 - 8 - 8 x -6 -16 -14 -12 -10 -8 -6 -4 Notice the bold section of the line and that the circle on the negative six is filled in. Let’s graph this answer on a number line. 2x + 3 13 2x 10 - 3 - 3 x 5 3 5 7 9 11 13 15 Notice the bold section of the line and that the circle on the five is filled in. 9x - 1 < 5x + 7 - 5x - 5x 4x - 1 < 7 + 1 + 1 4x < 8 x < 2 -6 -4 -2 0 2 4 Notice the bold section of the line and that the circle on the two is open.

4. Absolute Value Inequalities That means the distance from zero is less than 2 units. -3 -2 -1 0 1 2 3 x is between the points. That means the distance from zero is more than 2 units. -3 -2 -1 0 1 2 3 x is split on either side of the points. If then Between If then or Split

5. Solving Absolute Value Inequalities Less than means between Remove the absolute value symbol and write the equation according to the rules. Solve both sides of the inequality at the same time. Write your answer in set form Graph your solution on a number line. -4 -3 -2 -1 0 1 2 3 4 5 That was easy

6. More Solving Absolute Value Inequalities Greater than means split Remove the absolute value symbol and write the equation according to the rules. Solve both inequalities. Write your answer in set form Graph your solution on a number line. -4 -3 -2 -1 0 1 2 3 4 5 That was easy

7. Absolute Value Word Problems A rocket is shot into the air with an initial velocity of 232 feet per second. Since speed is the absolute value of velocity, the rocket’s speed, s, in feet per second after t seconds is Find the values of t for which s is less than 56 feet per second. I’d rather stick a pencil in my eye than try to solve this equation. Don’t do that. Let me show you how easy it is. Don’t forget to reverse the inequality symbol when you divide by a negative. 1 2 3 4 5 6 7 8 9 10

8. Quadratic Inequalities Quadratic Inequalities can be solved algebraically of graphically. Algebraically Solve by factoring first. -1 -2 -1 0 1 2 3 4 5 6 Graphically Enter the equation into your calculator The graph intersects the x-axis at -1 and 3, and all points between these two values are below the x-axis.

9. More Quadratic Inequalities Algebraically -5 -4 -3 -2 -1 0 1 2 3 Graphically The graph intersects the x-axis at -4 and 2, and all points between these two values are above the x-axis. Reverse the inequality symbol when you divide by a negative. Greater than means split.

10. General Rules for Quadratic Inequalities Quadratic InequalitySolution IntervalGraph of Solution r represents the smaller root, and R represents the larger root. r R