~ Chapter 3 ~
Solving Inequalities
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
Inequalities & Their Graph
Cumulative Review Chap. 1-2
Inequalities & Their Graph
Extra Practice - Chap 2
Inequalities & Their Graph
Extra Practice - Chap 2
Inequalities & Their Graph
Extra Practice - Chap 2
Inequalities & Their Graph
Notes
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
Notes
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 3-1
Solving Inequalities Using Addition &
Subtraction
Practice 3-1
Solving Inequalities Using
Addition & Subtraction
Notes
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
-7
-7
x > 15
Graph the solutions
m – 5 < -61
+5
m
+5
< -56
Solving Inequalities Using
Addition & Subtraction
Notes
-3.3 ≤ x + 7.5
-7.5
-7.5
-10.8 ≤ x
t - 5 ≥ 11
+5
+5
t ≥ 16
Graph the solutions
Homework – Practice 3-2 # 1-20
Solving Inequalities Using
Addition & Subtraction
Practice 3-2
Solving Inequalities Using Addition & Subtraction
Practice 3-2
Solving Inequalities Using
Addition & Subtraction
Notes
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
Solving Inequalities Using Mult. & Div.
Practice 3-2
Solving Inequalities Using Mult. & Div.
Notes
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
÷11
÷11
x > 2
Graph the solutions
m/5 < -6
x5
m
x5
< -30
Solving Inequalities Using Mult. & Div.
Notes
-¾ b ≤ 3
x(-4/3)
b ≥
-4.5 ≤ -0.9 p
x(-4/3)
-4
÷(-0.9)
÷(-0.9)
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 3-3
Solving Multi-Step Inequalities
Practice 3-3
Solving Multi-Step Inequalities
Practice 3-3
Solving Multi-Step Inequalities
Notes 3-4
Solving inequalities with variables on one side
-3x -4 ≤ 14
+4
+4
-3x ≤ 18
÷ (-3) ÷(-3)(reverse)
x ≥ -6
5 < 7 – 2t
-7
-7
-2 < -2t
÷(-2) ÷(-2) (reverse)
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
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
b>3
-6(x – 4) ≥ 7(2x – 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 Inequalities
Practice 3-4
Compound Inequalities
Practice 3-4
Compound Inequalities
Notes
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
-2 ≤ x
and
3x < 15
x<5
Compound Inequalities
Notes
7 < -3n + 1 ≤ 13
7 < -3n + 1
Solve & Graph…
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
x<2
or
Graph the solution
x≥3
Homework Practice 3-5 odd
Absolute Value Equations & Inequalities
Practice 3-5
Absolute Value Equations & Inequalities
Practice 3-5
Absolute Value Equations & Inequalities
Notes
Absolute Value – distance a number is away from 0.
Solving an absolute value equation
|x| + 5 = 11
-5
-5
|x| = 6
x = 6 & x = -6
3|n| = 15
|t| - 2 = -1
+2
+2
|t| = 1
t = 1 & t = -1
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
Notes
|c - 2| = 6
this means… c - 2 = 6
Solve …
c=8
Your turn…
-5.5 = |r + 2|
-3|y - 3| = 9
or c - 2 = -6
c = -4
Why?
Absolute Value Equations & Inequalities
Notes
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 3-6
Absolute Value Equations & Inequalities
Notes
Solve & graph the solutions…
v – 3 ≤ -4
or
|v - 3| ≥ 4 Rule 1 or 2?
v–3≥4
v ≤ -1
v≥7
Solve & graph the solutions…
-5 < w + 2 < 5
-5 < w + 2
-7 < w
|w + 2| < 5
Rule 1 or 2?
and
w+2<5
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 3-6
Absolute Value Equations & Inequalities
Practice 3-6
~ Chapter 3 ~
Chapter Review
~ Chapter 3 ~
Chapter Review