~ Chapter 3 ~Solving

Inequalities

Algebra I

Lesson 3-1 Inequalities & Their Graph

Lesson 3-2 Solving Inequalities Using Addition & Subtraction

Lesson 3-3 Solving Inequalities Using Mult. & Div.

Lesson 3-4 Solving Multi-Step Inequalities

Lesson 3-5 Compound Inequalities

Lesson 3-6 Absolute Value Equations & Inequalities

Chapter Review

Algebra I

Inequalities & Their Graph

Cumulative Review Chap. 1-2

Lesson 3-1

Inequalities & Their Graph

Extra Practice - Chap 2Lesson 3-1

Inequalities & Their GraphExtra Practice - Chap 2Lesson 3-1

Inequalities & Their GraphExtra Practice - Chap 2Lesson 3-1

Inequalities & Their Graph

NotesLesson 3-1

Inequality – mathematical sentence that contains >, <, ≥, ≤, ≠.Solution of an inequality – any number that makes the inequality true.Example ~ y > 5 Would 6 make the inequality true? What about 7, 4, 9, 22, -5?Graphing solutions of Inequalities

Graph y < 3

Graph x > -1

Graph a ≤ -2

Graph -6 ≤ g

Inequalities & Their Graph

NotesLesson 3-1

Graph ½ ≥ c

Writing Inequalities to Describe Graphs…

Define a variable and write an inequality

A bus can seat at most 48 students

You must be at least 16 years old to obtain a driver’s license

Homework – Practice 3-1

Solving Inequalities Using Addition & Subtraction

Practice 3-1Lesson 3-2

Solving Inequalities Using Addition & Subtraction

Practice 3-1Lesson 3-2

Solving Inequalities Using Addition & Subtraction

NotesLesson 3-2

Equivalent Inequalities – Inequalities with the same solution. Ex ~ x > 2 and x – 5 > -3

Steps for solving an inequality

(1) Divide the equation at the inequality sign into two equal sides

(2) Underline the variable.

(3) Identify the number on the same side as the variable.

(4) Identify the operation (addition or subtraction) and perform the opposite (inverse operation) to both sides of the equation.

(5) Simplify and solve the inequality

(6) Check your answer.

x + 7 > 22 m – 5 < -61

-7 -7 +5 +5

x > 15 m < -56

Graph the solutions

Solving Inequalities Using Addition & Subtraction

NotesLesson 3-2

-3.3 ≤ x + 7.5 t - 5 ≥ 11

-7.5 -7.5 +5 +5

-10.8 ≤ x t ≥ 16

Graph the solutions

Homework – Practice 3-2 # 1-20

Solving Inequalities Using Addition & Subtraction Practice 3-2Lesson 3-2

Solving Inequalities Using Addition &

Subtraction Practice 3-2

Lesson 3-2

Solving Inequalities Using Addition & Subtraction

NotesLesson 3-2

Write & Solve the Inequality that models each situation.

Your baseball team has a goal to collect at least 160 blankets for a shelter. Team members brought 42 blankets on Monday and 65 blankets on Wednesday. How many blankets must the team donate on Friday to make or exceed their goal?

Your brother has $2000 saved for a vacation. His airplane ticket is $637. How much can he spend for everything else?

Homework – Practice 3-2 #21-42

Solving Inequalities Using Mult. & Div. Practice 3-2

Lesson 3-3

Solving Inequalities Using Mult. & Div. Practice 3-2

Lesson 3-3

Solving Inequalities Using Mult. & Div. Notes

Lesson 3-3

Steps for solving an inequality

(1) Divide the equation at the inequality sign into two equal sides

(2) Identify the variable.

(3) Identify the number on the same side as the variable.

(4) Identify the operation (multiplication or division) and perform the opposite (inverse operation) to both sides of the inequality.

(5) Simplify and solve the inequality

(If you multiply or divide each side of an inequality by a negative number, you reverse the inequality symbol. )

(6) Check your answer.

11x > 22 m/5 < -6

÷11 ÷11 x5 x5

x > 2 m < -30

Graph the solutions

Solving Inequalities Using Mult. & Div. Notes

Lesson 3-3

-¾ b ≤ 3 -4.5 ≤ -0.9 p

x(-4/3) x(-4/3) ÷(-0.9) ÷(-0.9)

b ≥ -4 5 ≥ p (another way to solve?)

Graph the solution

Write & Solve the Inequality that models each situation.

Students in the school band are selling calendars. They earn $0.40 on each calendar they sell. Their goal is to earn more than $327. Write and solve an inequality to find the fewest number of calendars they can sell and still reach their goal.

Suppose you earn $8.15 per hour working part time at the dry cleaner. Write and solve an inequality to find how many full hours you must work to earn at least $100.

Homework ~ Practice 3-3 odd

Solving Multi-Step Inequalities Practice 3-3Lesson 3-4

Solving Multi-Step Inequalities Practice 3-3

Lesson 3-4

Solving Multi-Step Inequalities Practice 3-3

Lesson 3-4

Solving Multi-Step Inequalities Notes 3-4

Lesson 3-4

Solving inequalities with variables on one side

-3x -4 ≤ 14 5 < 7 – 2t

+4 +4 -7 -7

-3x ≤ 18 -2 < -2t

÷ (-3) ÷(-3)(reverse) ÷(-2) ÷(-2) (reverse)

x ≥ -6 1 > t or t < 1

Now you solve some…

-8 < 5n – 23 12 – 5k ≤ 2

3 < n k ≥ 2

Distributive Property & Inequalities

4p + 2(p + 7) < 8 15 ≤ 5 – 2(4m + 7)

4p + 2p + 14 < 8 15 ≤ 5 – 8 m – 14

6p + 14 < 8 15 ≤ -8m - 9

Then… Solve like other multi step inequality

Solving Multi-Step Inequalities Notes

Lesson 3-4

Solving inequalities with variables on both sides

6z – 15 < 4z + 11 3(4 – m) ≥ 4(2m + 1)

z < 13 m ≤ 8/11

Your turn…

3b + 12 > 27 – 2b -6(x – 4) ≥ 7(2x – 3)

b > 3 2.25 ≥ x or 2 ¼ ≥ x

Write & Solve an inequality

One half the difference of t and six is less than or equal to four

½(t – 6) ≤ 4

The perimeter of an isosceles triangle is at most 27 cm. One side is 8 cm long. Find the possible length of the two congruent sides.

Homework Practice 3-4 odd

Compound InequalitiesPractice 3-4Lesson 3-5

Compound Inequalities

Practice 3-4Lesson 3-5

Compound Inequalities

NotesLesson 3-5

Compound Inequalities – Two inequalities joined by the word and or or.

For example x > -6 and x < 8… How could we write this?

-6 < x < 8 Graph?

The solution for “and” joined inequalities is the overlap of the two graphs… i.e. where both graphs show the same solutions.

Write and graph the compound inequalitity

All real numbers greater than -2 but less than 9

-2 < x < 9

The books were priced between $3.50 and $6.00, inclusive.

3.50 ≤ c ≤ 6.00

Solving a compound inequality containing and…

Solve each inequality… then simplify

-6 ≤ 3x < 15 solve -6 ≤ 3x and 3x < 15

-2 ≤ x x < 5

Compound Inequalities

NotesLesson 3-5

7 < -3n + 1 ≤ 13 Solve & Graph…

7 < -3n + 1 and -3n + 1 ≤ 13 Solution: -2 > n and n ≥ -4 or -4 ≤ n < -2

Writing compound Inequalities with or

Discounted fares are available to children 12 and under or to adults at least 60 years of age.

a ≤ 12 or a ≥ 60 Graph the solution…

What else do we know?

Write an inequality that represents all real numbers that are at most -5 or at least 3. Graph your solution.

Solving a compound inequality containing or

-2x + 7 > 3 or 3x – 4 ≥ 5 Graph the solution

x < 2 or x ≥ 3

Homework Practice 3-5 odd

Absolute Value Equations & Inequalities

Practice 3-5Lesson 3-6

Absolute Value Equations & Inequalities

Practice 3-5Lesson 3-6

Absolute Value Equations & Inequalities

NotesLesson 3-6

Absolute Value – distance a number is away from 0.

Solving an absolute value equation

|x| + 5 = 11 |t| - 2 = -1

-5 -5 +2 +2

|x| = 6 |t| = 1

x = 6 & x = -6 t = 1 & t = -1

3|n| = 15 4 = 3|w| - 2

|n| = 5 2 = |w|

n = 5 & n = -5 w = 2 & w = -2

More absolute value equations

Sometimes an absolute value equation has the expression inside the absolute value symbols.

Solving Absolute Value Equations ~ To solve an equation in the form |A| = b, where A represents a variable expression and b > 0, solve A = b and A = -b.

Absolute Value Equations & Inequalities

NotesLesson 3-6

|c - 2| = 6 this means… c - 2 = 6 or c - 2 = -6 Why?

Solve … c = 8 c = -4

Your turn…

-5.5 = |r + 2| -3|y - 3| = 9

Absolute Value Equations & Inequalities

NotesLesson 3-6

Solving Absolute Value Inequalities

|n - 2| < 5 (represents all numbers whose distance from 2 is less than 5 units)

So… -5 < n - 2 < 5 Graph the solution

|n - 2| > 5 (represents all numbers whose distance from 2 is more than 5 units)

So… n – 2 < -5 or n-2 > 5 Graph the solution

Here are the rules…

Rule 1 ~ To solve an inequality in the form |A| < b, where A is a variable expression and b > 0, solve –b < A < b.

Rule 2 ~ To solve an inequality in form |A| > b, where A is a variable expression and b > 0, solve A < -b or A > b.

Similar rules are true for |A| ≤ b or |A| ≥ b.

Absolute Value Equations & Inequalities

Practice 3-6Lesson 3-6

Absolute Value Equations & Inequalities

NotesLesson 3-6

Solve & graph the solutions… |v - 3| ≥ 4 Rule 1 or 2?

v – 3 ≤ -4 or v – 3 ≥ 4

v ≤ -1 v ≥ 7

Solve & graph the solutions… |w + 2| < 5 Rule 1 or 2?

-5 < w + 2 < 5 -5 < w + 2 and w + 2 < 5

-7 < w and w < 3 … -7 < w < 3 graph…

Write an absolute value inequality and solve

All numbers less than 3 units from 0

|n| < 3

The ideal diameter of a gear for a certain type of clock is 12.24 mm. An actual diameter can vary by 0.06 mm. Find the range of acceptable diameters.

|d – 12.24| ≤ 0.06 Rule 1 or 2?

-0.06 ≤ d – 12.24 ≤ 0.06

Homework ~ Practice 3-6

#1-28 even & 29-36

Absolute Value Equations & Inequalities

Practice 3-6Lesson 3-6

Absolute Value Equations & Inequalities

Practice 3-6Lesson 3-6

~ Chapter 3 ~Chapter Review

Algebra I Algebra I

~ Chapter 3 ~Chapter Review

Algebra I Algebra I