6.1 and 6.2 Solving Inequalities

Definitions Graph of an inequality---the set of all points on a number line which represents all solutions Solution of an inequality—value which makes the inequality TRUE

Let’s compare the graphs of the two inequalities

Closed circle Open circle If the endpoint is not a solution, use an open circle at that point. < or > If the endpoint is a solution, use a shaded in circle at the point. ≤ or ≥

Investigation Write the following inequality Do each of the following steps by beginning with the original inequality at each step and analyze how it affects the inequality: Add any number to both sides of the inequality Subtract any number from both sides of the inequality Multiply or divide both sides by a positive number Multiply or divide both sides by a negative number

Conclusion: If you multiply or divide each side by a negative number, reverse the inequality.

a) Solve -8+n≥-9 -8+n≥-9 n≥-1 +8 +8 n≥-1 All numbers greater than OR equal to -1

b) Solve -7x<-21 -7x<-21 -7 -7 x>3

Let’s Try to Check You can’t CHECK all the solutions of an inequality, so choose several SOLUTIONS and check those. Some in the shaded area, some not in the shaded area, and the “critical” point.

c) Solve 3(x+2)≤7 3x+6≤7 -6 -6 3x≤1 3x≤1 3 3 x≤1/3

d) Solve -3(x-3)>6 -3x+9>6 -9 -9 -3x>-3 -3x>-3 -3 -3 Don’t Forget—Reverse the inequality if multiplying or dividing by a negative number -3x+9>6 -9 -9 -3x>-3 -3x>-3 -3 -3 x<1

e) Solve 2x-3≥4x-1 2x-3≥4x-1 -2x -2x -3≥2x-1 +1 +1 -2≥2x 2 2 -1≥x x≤-1

2x+12≥7x+7 -2x -2x 12≥5x+7 -7 -7 5≥5x 5 5 1≥x x≤1 f) Solve 2x+12≥7(x+1) 2x+12≥7x+7 -2x -2x 12≥5x+7 -7 -7 5≥5x 5 5 1≥x x≤1

How many amusement rides? An amusement park charges $5 for admission and $1.25 for each ride. You go to the park with $25. Write an inequality that represents the possible number of rides you can go on. What is the maximum number of rides you can go on? x= max number of rides 1.25x + 5 ≤ 25 x ≤ 16; You can ride 16 or less rides!!!

How many toppings? You have $18.50 to spend on pizza. It costs $14 plus $0.75 for each additional topping (tax is included). Solve an inequality to find the maximum number of toppings the pizza can have. x= max number of toppings .75x + 14 ≤ 18.50 x ≤ 6; You can have 6 or less toppings!!!

Assessment Card Create a real life example involving an inequality in which the following graph would be its solution.

Which graph shows the solution to 2x - 10 ≥ 4?

Solve -2x + 6 ≥ 3x - 4 a) x ≥ -2 b) x ≤ -2 c) x ≥ 2 d) x ≤ 2

What are the values of x if 3(x + 4) - 5(x - 1) < 5? a) x < -6 b) x > -6 c) x < 6 d) x > 6

THE END!!