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

HW#2:Solving Multi-Step Inequalities 1st, Tues, 2/7 SWBAT… solve inequalities using addition, subtraction, multiplication, division Agenda WU (10 min) Review HW#1: Absolute value inequalities (10 min) Solving inequalities – 8 examples (25 min) Warm-Up: Solve for n: 3.7| n | + 10.6 = 3.2 Solve for x: 7.25| x + 1 | + 6.8 = 21.3 HW#2:Solving Multi-Step Inequalities No Solution x = 1 or x = -3 8 8

Solving and Graphing Inequalities by Addition and Subtraction

Directions: Solve the inequality, graph the solution on a number line. Ex 1: d – 14 ≥ -19 Ex 2: 22 > m – 8 Ex 3: Three added to a number is no more 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 3, so to write the expression with x on the left, use x > 3

Solving and Graphing Inequalities by Multiplication and Division

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

Ex 1: -7d ≤ 147 Ex 2: 5n ≤ -25 Ex 3: Ex 4: Ex 5: Directions: Solve the inequality, graph the solution on a number line. Ex 1: -7d ≤ 147 Ex 2: 5n ≤ -25 Ex 3: Ex 4: Ex 5: d ≥ -21 [-21, ∞) n ≤ -5 (Do NOT change the inequality sign) r > -49 (-49, ∞) x > -18 (Do NOT change the inequality sign) (-18, ∞) n < -48 (-∞, -48)

Solving and Graphing Multi-Step Inequalities

Two times the difference of a number and five is no more than eight. Tues, 2/7 SWBAT… solve multi-step inequalities Agenda: Five WU problems below (15 min) Double math courses (5 min) Review HW#2 – multi-step inequalities (10 min) Review Quiz (5 min) Electra’s truck problem (10 min) WU: Solve each inequality & write in interval notation: 6(x – 11) – 4x ≤ -72 Two times the difference of a number and five is no more than eight. -7(k + 4) + 11k ≥ 8k – 2(2k + 1) 2(4r + 3) ≤ 22 + 8(r – 2) 1.) x ≤ -3 (-∞, -3] 2.) n = number; 2(n – 5 ) ≤ 8; n ≤ 9 [9, ∞) 3.) Empty set 4.) All real numbers 17 17

Geometry AND Honors Advanced Algebra with Trigonometry (Algebra II) Infinity Math Sequence for students that take Geometry and Honors Advanced Algebra in 10th grade: 9th 10th 11th 12th *Honors or Regular Algebra *Honors or Regular Geometry and Honors Advanced Algebra *Honors Pre-Calculus *AP Statistics (college credit ≥ 4) (optional) *AP Calculus (college credit ≥ 4) *AP Statistics (college credit ≥ 4) (optional)

Electra needs to rent a truck for a day to move some furniture Electra 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 her home. a.) Write an inequality that Electra can use to find the maximum number of miles that she 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 Electra can drive so that she spends less than she 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.) 29.95 + 0.87m < 72.00 where m is the number of miles driven. 29.95 + 0.87m < 72.00 0.87m < 42.05 m < 48.333 or 48 1/3 miles 19

Thurs, 2/9 SWBAT… solve compound inequalities Agenda WU (5 min) Solving compound inequalities – 6 examples (25 min) Two open ended compound inequalities (10 min) Work on HW2 or HW3 (10 min) Warm-Up: -(3t – 5) + 7 > 8t + 3 HW#3:Solving Compound Inequalities 1.) 20 20

Solving and Graphing Compound Inequalities

h ≥ 52 and 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 and h ≤ 72 The inequalities h ≥ 52 and h ≤ 72 can be combined and written without using “and” as 52 ≤ h ≤ 72 Graph the inequality “sandwich” Variable is isolated

You try! Inequalities Containing and Solve and graph the compound inequality. Write it two different ways. 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 Graph the inequality “torpedo”

You try! Inequalities Containing or Solve and graph the compound inequality. 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

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

Tues, 2/ SWBAT… solve compound inequalities Agenda WU (10 min) Review HW3 and HW#4 (20 min) Solve the inequality: 1.) Solve for a: 12 – (a + 3) > 4a – (a – 1) HW4: Solving Compound Inequalities: Chemistry & Geometry 1.) a < 2 (did you forget to flip the sign?!?  ) If not,  27

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.

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