Graphing and Transforming Inequalities When graphing a one variable inequality, there are specific ways to demonstrate the concepts of >, <, >, < | | | | | | | | | | | | | | | | | | | | | | -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Transformations that Produce Equivalent Inequalities Operation Original Equivalent Add the same number x – 3 < 5 Add 3 x < 8 Subtract same number x + 6 > 10 Subtract 6 x > 4 Multiply by same + number ½ x > 3 Multiply by 2 x > 6 Divide by same + number 3x < 9 Divide by 3 x < 3 Multiply by same Negative # -x < 4 Multiply by -1 x > -4 * Reverse the inequality sign Divide by same Negative # -2x < 6 Divide by -2 x > -3

Investigating Inequalities – Impact of Operations Take the basic example of 4 > -8 Does the inequality sign change direction? A positive number A negative number Add Subtract Multiply Divide No No No Yes What about when we are dealing with a variable?

Compound Inequalities Compound inequalities are used to express a value that is “bounded” or unlimited in opposite directions. They are linked by the words “and” or “or” Compound Inequalities are graphed using the same principles of solid and open dots to show whether or not the solution set includes the specific value at the end point. | | | | | | | | | | | | | | | | | | | | | | -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 1. All real numbers that are greater than zero and less than or equal to 4 | | | | | | | | | | | | | | | | | | | | | | -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 2. All real numbers that are less than -1 or greater than 2

| | | | | | | | | | | | | | | | | | | | | | Solving Compound Inequality with “And,” “Or,” and Reversing both Symbols 1. Solve -2 < 3x – 8 < 10 | | | | | | | | | | | | | | | | | | | | | | -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 2. Solve 3x + 1 < 4 or 2x – 5 > 7 | | | | | | | | | | | | | | | | | | | | | | -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 3. Solve -2 < -2 – x < 1 | | | | | | | | | | | | | | | | | | | | | | -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Using the concepts of Compound Inequalities Letting y represent the approximate elevation (in feet), write an inequality to describe the regions of Mount Rainier. Timber Region below 6000 feet, down to 2000 feet. Mount Rainier 14,400 ft_______________________________________________ 7500 ft ________________________________________________ 6000 ft________________________________________________ 2000 ft________________________________________________ b. Alpine meadow region below 7500 feet. c. Glacier and permanent snow field region

Solving an Absolute Value Equation Solve |x – 2| = 5 The equation has two solutions, the expression x -2 can therefore be equal to either 5 or -5. In order to solve the equation, we actually solve for both options x – 2 is positive x – 2 is negative |x – 2| = 5 |x – 2| = 5 x – 2 = + 5 x – 2 = -5 x = 7 x = 3 Solve |2x – 7| - 5 = 4 Again, there are two solutions, both a positive and a negative. We also need to isolate the absolute value expression on one side of the equation. 2x – 7 is positive 2x – 7 is negative |2x – 7| - 5 = 4 |2x – 7| - 5 = 4 |2x – 7| = 9 |2x – 7| = 9 2x – 7 = +9 2x – 7 = - 9 2x = 16 2x = -2 X = 8 x = -1

Solving Absolute Value Equations and Inequalities Each Absolute Value Inequality can be rewritten as two equations or two inequalities joined by “and” or “or.” |ax + b| < c means ax + b < c and ax + b > -c |ax + b| = c means ax + b = c or ax + b = -c |ax + b| > c means ax + b > c or ax + b < -c |ax + b| > c means ax + b > c or ax + b > -c

Solving Absolute Value Inequalities Solve |x – 4| < 3 We still solve for x – 4 as both a positive and a negative, but when solving for the negative value WE MUST REVERSE the Inequality Sign. x – 4 is positive x – 4 is negative |x – 4| < 3 |x – 4| < 3 x – 4 < + 3 x – 4 > -3 x < 7 x > 1  Reverse Inequality Solve |2x + 1| - 3 > 6 2x + 1 is a positive 2x + 1 is negative |2x + 1| - 3 > 6 |2x + 1| - 3 > 6 |2x + 1| > 9 |2x + 1| > 9 2x + 1 > 9 2x + 1 < -9 2x > 8 2x < -10 x > 4 x < -5  Reverse Inequality

Graphing Inequalities in Two Variables Forms of the Inequality: ax + by < c ax + by > c Determining if an ordered pair is a Solution Substitute the values for x and y For the inequality 3x – 3y > -2, is the following ordered pair a solution? c. (2, -1) a. (0, 0) b. (0, 1)

Graphing Linear Inequalities Graphing a Linear equality How would we graph (in a coordinate plane) x < -2?

Graphing Linear Inequalities Graphing a Linear equality How would we graph (in a coordinate plane) y < 1?

Graphing Linear Inequalities Graphing a Linear equality How would we graph x + y > 3? y = -x + 3

Graphing Linear Inequalities Graphing a Linear equality How would we graph 2x - 3y > -2? y < 2/3 x + 2/3

Working with Inequalities You’re listening to the basketball game on the car radio. At half-time La Salle has already scored 24 points, but you have to turn off the car radio and go to work. Let x represent the number of 2 point baskets and y represent the number of 3 point baskets scored. Write and graph the inequality that describes the different number of 2 point and 3 point shots that La Salle could have scored by the end of the game.

You have $125 to spend on school clothes You have $125 to spend on school clothes. It costs $20 for a pair of pants and $15 for a shirt. Let x represent the number of pants you can buy and let y represent the number of shirts you can buy. Write and graph an inequality that describes the different number of pants and shirts you can buy

The area of a window shown at the right is less than 28 square feet The area of a window shown at the right is less than 28 square feet. Let x and y represent the height of the triangle and rectangular portions of the window. Write and graph an inequality that describes the different dimensions of the window. x __ | y |----------4--------------|

Chapter Review

Which inequality is equivalent to -5x + 4 < -2x + 7?

2. You are at a used book sale. Soft covers are $0 2. You are at a used book sale. Soft covers are $0.75 each and hard covers are $1.50 each. If you have $6.00 to spend and you buy 4 soft covers, how many hard covers can you buy? a. 0 b. 1 c. 2 d. 3

3. Which inequality represents the statement “x is less than 5 and at least -5?” a. -5 < x < 5 b. -5 < x < 5 c. -5 < x < 5 d. -5 < x < 5

4. Solve -23 < 3x – 2 < 13. a. -7 < x < 5 b. -25/3 < x < 11/3 c. -7 < x < 5 d. -25/3 < x < 11/3

5. Graph the solution to 6x – 4 > 14 or 3x + 10 < 4.

6. Solve |8x = 2| - 4 = 18. a. 3/2 and -2 b. -2 and -3 c. -3 and 5/2 d. -3/2 and 2

7. Which ordered pair is not a solution for 5x + 4y < -12? b. (-2, 4) c. (-4, 0) d. (-3, -8)

8. Choose the inequality whose solution is shown in the graph. a. 3y – 4x > 4 b. 4x – 3y > 4 c. 3y – 4x < 4 d. 4x – 3y < 4