HW#2:Solving Inequalities Wed, 2/1 SWBAT… solve inequalities using addition, subtraction, multiplication, division Agenda WU (5 min) Review HW#1 (10 min) Inequalities charts (10 min) Solving inequalities – 8 examples (20 min) Warm-Up: 1. Take out HW#1 2. Set up notes: Topic = Solving inequalities HW#2:Solving Inequalities 1

Phrases for Inequalities < > ≤ ≥

Phrases for Inequalities < > ≤ ≥ less than fewer than greater than more than at most no more than less than or equal to at least no less than greater than or equal to

Endpoints (when graphing on a number line) < > ≤ ≥

Endpoints (when graphing on a number line) < > ≤ ≥ Open Circle Closed circle

x > 4 (4, ∞ ) -2 ≤ x ≤ 5 [-2, 5] -2 < x ≤ 5 (-2, 5] Interval Notation < > ≤ ≥ ( ) [ ] Examples: x > 4 (4, ∞ ) -2 ≤ x ≤ 5 [-2, 5] -2 < x ≤ 5 (-2, 5] 0 4 -2 0 5 -2 0 5

Always use parenthesis with Infinity Interval Notation < > ≤ ≥ ( ) [ ] Examples: x ≥ 4 or x < 1 (-∞, 1) [4, ∞) Always use parenthesis with Infinity 0 1 4

Solving Inequalities by Addition and Subtraction

Directions: Solve the inequality, graph the solution on a number line, and write in interval notation. Ex 1: d – 14 ≥ -19 Ex 2: 22 > m – 8 Ex 3: Three more than a number is less than twice the number. 1.) d ≥ -5 [-5, ∞) 2.) m < 30 (-∞, 30) 3.) Let x = a number 3 + x < 2x (subtract n from both sides) 3 < x x > 3 (3, ∞)

WARNING!!!!! (Example 2 & 3) An equation such as x = 5 can be written as 5 = x (because of the Symmetric Property of Equality) You CANNOT rewrite an inequality such as 3 < x as x < 3. The inequality sign always points to the lesser value (or it’s eating the bigger number.) In 3 < x, the inequality points to n, so to write the expression with y on the left, use x > 3.

Solving Inequalities by Multiplication and Division

Ex 1: -7d ≤ 147 Ex 2: Ex 3: Example 4: Directions: Solve the inequality, graph the solution on a number line, and write in interval notation. Ex 1: -7d ≤ 147 Ex 2: Ex 3: Example 4: d ≥ -21 [-21, ∞) r > -49 (-49, ∞) x > -18 (Do change the inequality sign) (-18, ∞) n < -48 (-∞, -48)

Very important…. <  > > < ≤  ≥ ≥  ≤ When you multiply or divide each side of an inequality by a negative number… you always reverse or flip the inequality sign. <  > > < ≤  ≥ ≥  ≤

RATIONALE -14 < -8 7 > 4 -14 > -8 7(-2) > 4(-2) NOT TRUE! You must change the inequality symbol -14 < -8

Exit Slip: Solve, graph, write in interval notation 4(3t – 5) + 7 > 8t + 3 Answer: t > 4 (4, ∞)

Solve each inequality: 1 Solve each inequality: 1.) Two times the difference of a number and five is greater than eight. 2.) -7(k + 4) + 11k ≥ 8k – 2(2k + 1) 3.) 2(4r + 3) ≤ 22 + 8(r – 2) 1.) n = number; 2(n – 5 ) > 8; n > 9 (9, ∞) 2.) Empty set 3.) All real numbers 16 16

Solve each inequality: 1. ) 3(x – 4) < -15 2 Solve each inequality: 1.) 3(x – 4) < -15 2.) 6(x – 11) – 4x ≤ -72 3.) 4(3t – 5) + 7 > 8t + 3 4.) Two times the difference of a number and five is greater than eight. 5.) -7(k + 4) + 11k ≥ 8k – 2(2k + 1) 6.) 2(4r + 3) ≤ 22 + 8(r – 2) 1.) x < -1 (-∞, -1) 2.) x ≤ -3 (-∞, -3] 3.) t > 4 (4, ∞) 4.) n = number; 2(n – 5 ) > 8; n > 9 (9, ∞) 5.) Empty set 6.) All real numbers 17

Thurs, 2/2 SWBAT… solve compound inequalities Agenda WU (10 min) Compound inequalities examples (30 min) Warm-Up: Bruce needs to rent a truck for a day to move some furniture. The table below shows the rates of the two truck-rental companies near his home. a.) Write an inequality that Bruce can use to find the maximum number of miles that he can drive and spend less with Company A than Company B. Be sure to identify your variable or variables. b) Find the maximum number of miles that Bruce can drive so that he spends less than he would for a truck rented from Company B. Company Daily rate Per mile charge A $29.95 $0.87 B $72.00 $0.00 a.) $0.87m + $29.95 < $72.00 where m is the number of miles driven. $0.87m + $29.95 < $72.00 $0.87 < 42.05 m < 48.333 or 48 1/3 miles 18

Solving Compound Inequalities

h ≥ 52 h ≤ 72 Inequalities Containing and To ride a roller coaster, you must be at least 52 inches tall, and your height cannot exceed 72 inches. If h represents the height of the rider, we can write two inequalities to represent this. At least 52 inches Cannot exceed 72 inches h ≥ 52 h ≤ 72 The inequalities h ≥ 52 and h ≤ 72 can be combined and written without using “and” as 52 ≤ h ≤ 72 Graph the inequality

You try! Solve the compound inequality and graph the solution set. 1.) -2 < x – 3 < 4 2.) -5 < 3p + 7 ≤ 22 1.) 1 < x < 7 2.) -4 < p ≤ 5

Inequalities Containing or SNAKES Most snakes live where the temperature ranges from 750 F to 900 F. Write an inequality to represent temperatures where snakes will not thrive. Let t = temperature t < 75 or t > 90

You try! Solve the compound inequality and graph the solution set. 5n – 1 < -16 or -3n – 1 < 8 The product of -5 and a number is greater than 35 or less than 10. n < -3 or n > -3 -5n > 35 or -5n < 10 n < -7 or n > -2

Tues, 2/ SWBAT… solve compound inequalities Agenda WU (10 min) Review hw#4 (20 min) Chemistry problem (10 min) Solve each inequality: 1.) Solve for a: 12 – (a + 3) > 4a – (a – 1) Chemistry Problem: pH levels of swimming pool 3.) a < 2 24

OPEN ENDED 1: Write a compound inequality containing and for which the graph is the empty set. Sample answer: x ≤ -4 and x ≥ 1 OPEN ENDED 2: Create an example of a compound inequality containing or that has infinitely many solutions. Sample answer: x ≤ 5 or x ≥ 1

Review hw#3: Compound inequalities

GEOMETRY The Triangle Inequality Theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side. a.) Write and solve three inequalities to express the relationships among the measures of the sides of the triangle shown above. b.) What are the possible lengths for the third side of the triangle? c.) Write a compound inequality for the possible values of x. 9 x 4 a.) x + 9 > 4, x > -5 x + 4 > 9, x > 5 4 + 9 > x, x < 13 b.) 6, 8, 7, 9, 10, 11, 12 c.) 5 < x < 13

Chemistry The acidity of the water in a swimming pool is considered normal if the average of three pH readings is between 7.2 and 7.8. The first two readings for the swimming pool are 7.4 and 7.9. What possible values for the third reading p will make the average pH normal? 7.2 ≤ (7.4 + 7.9 + p)/3 ≤ 7.8 3(7.2) ≤ 3(7.4 + 7.9 + p)/3 ≤ 3(7.8) 21.6 ≤ 15.3 + p ≤ 23.4 21.6 – 15.3 ≤ 15.3 + p – 15.3 ≤ 23.4 – 15.3 6.3 ≤ p ≤ 8.1 The value for the third reading must be between 6.3 and 8.1, inclusive.

The value for the third reading must be between 6. 3 and 8 The value for the third reading must be between 6.3 and 8.1, inclusive.