Inequalities and Algebra

1. Find three consecutive even integers whose sum is 210.

2. Five times a number increased by 40 is 175. Find the number.

3. The sum of two numbers is 30. The second number is 6 more than the first number. Find the numbers.

4. When 6 times a number is increased by 4 the result is 40. What

is the number?

5. One number exceeds another number by 5. If the sum of the two

numbers is 39. Find the smaller number.

INEQUALITIES

OBJECTIVES

1. Recall the symbols for inequalities

2. Translate a verbal statement into an inequality or vice versa

3. Identify the properties of inequality

A healthy person has normal platelet count of between 150,000 and 400,000 per cubic millimeter(mm3). If a person has a platelet count less than 50,000, continuous bleeding may occur which may lead to a life-threatening risk. Dengue fever is a life-threatening illness but can be treated if diagnosed properly. One way to confirm dengue is through a complete blood count. A person with this infection has a platelet count of less than 140,000/mm3 of blood and white blood cells count of less than 5,000/mm3

Platelets or thrombocytes are tiny colorless disk-shaped particles in the blood which is responsible for forming clots to stop bleeding from wounds.

A healthy person has normal platelet count of between 150,000 and 400,000 per cubic millimeter(mm3). If a person has a platelet count less than 50,000, continuous bleeding may occur which may lead to a life-threatening risk. Dengue fever is a life-threatening illness but can be treated if diagnosed properly. One way to confirm dengue is through a complete blood count. A person with this infection has a platelet count of less than 140,000/mm3 of blood and white blood cells count of less than 5,000/mm3

Recall: What is an equation?

An inequality is a mathematical

statement, which states that two

quantities are not equal.

Some examples of inequalities in everyday life

- SPEED LIMIT: 40KPH

s ≤ 40MAXIMUM Speed is 40KPH

Is a driver who is traveling at 40 kph driving at a legal

speed?

Children Under 5 – Free Entrance fee

c < 5

Can a child who is 5 years old enter for free?

Drink at least 8 glasses of water

g ≥ 8

Name three amounts of gifts that you can buy for the exchange gift?

Exchange Gift Spend between P300 to P500

P300 < X < P500

P300 < xX < P500

1. The speed is less than or equal to 40kph

2. The age is less than 5 years.

3. The number of glasses is greater than or equal to 8

s < 40

a < 5

g > 8

< > •less than•fewer than

•greater than•more than•exceeds

•less than or equal to•no more than•at most•maximum•cannot exceed

•greater than or equal to•no less than•at least•minimum

Inequality symbols

FYI: Thomas Harriot- introduced

the symbols < and >. Pierre Bouguer- first used ≤ and ≥ about a century later

Translate.

1. ten times m is not equal to one hundred.

2. Fifteen multiplied by p increased by two is less than or equal to twenty-eight.

3. Thrice y is greater than or equal to twelve and less than or equal to thirty.

Write an inequality for each sentence.

* Sandra’s grade in Math is greater than 90.

* Atleast 500 guests attended the Glee @ 50 Concert.

Let us now consider some

sentences that state inequalities.

1. Alicia’s grade in Math is lower

than Mariah’s grade. If x is the

grade of Alicia and y is the grade

of Mariah, then __________.x < y

2. St. Benedict’s building is

higher than St. Scholastica’s building.

If the height of St. Benedict’s

building is a and that of St.

Scholastica’s building is b, then

_________.a > b

3. We do not have an equal

number of students in Grade 7 and in Grade 8

If there are c Grade 7 students

and d grade 8 students, then

______.c d

Inequalities >, <, ≥, ≤, or ≠

What inequality symbol must be written on the blank to make each statement true?

1. If 10 __ 7 and 7 __ 5, then 10 __5.

2. a. If 8__ 6, then 8 +5 ___ 6 +5.

b. If 8__ 6, then 8 +(-2) ___ 6 +(-2).

3. a. If 25 __ 10, then 2(25) __2(10). b. If 25 __ 10, then ½ (25) __ ½ (10).

4. a. If 18 __ 12, then -3(18) __ -3(12)

b. If 18 __ 12, then -1/3 (18) __ -1/3 (12)

Trichotomy Axiom

For all real number a and b, one

and only one of the following is

true:

a < b a > b a = b

Transitive Axiom

For all real number a, b, and c:

If a < b and b < c, then a < c.

If a > b and b > c, then a > c.

Addition Property for Inequality

For all real numbers a, b, and c:

If a < b, then a + c < b + c.

If a > b, then a + c > b + c.

Multiplication Property for Inequality

For all real numbers a, b, and c:

If a < b, then ac < bc when c > 0.

If a > b, then ac < bc when c < 0.

Why is zero excluded in the statement of multiplication property for inequalities?

Translate each statement into an inequality

1. Manny Pacquiao must not weigh over 130

pounds for the lightweight title fight.

2. Gina prepared for her birthday;she baked over 6 dozen muffins.

3. The height of the tree is at least 20 meters.

4. The cost of an expensive coffee is at most

Php 70 per pound.

5. The discount is no less than Php 40 per

price.

What are the symbols used for inequalities?

< , > , , and

Enumerate the properties of

inequality?

Trichotomy AxiomTransitive AxiomAddition Property for InequalityMultiplication Property for

Inequality

Solutions and graphs of linear inequalities in One Variable

OBJECTIVES

Demonstrate understanding of linear inequalities in one variable;

Graph an inequality on a number line; and

Solve for linear inequalities in One Variable.

A. Identify the following symbols.

a. >

b. <

c.

d.

B. Find the value of x.

a. x + 4 = 7

b. 2x + 5 = x + 8

c. x > 3

x = 3

x = 3

x = 4, 5, 6,…

The graph of x = 3 is

How is the graph of x > 3 represented?

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Graph each inequality.

1. x ≥ –10

2. x < –3

Warm UpGraph each inequality. Write an inequality for each situation. 1. The temperature must be at least –10°F. 2. The temperature must be no more than 90°F.

x ≥ –10

x ≤ 90

Solve each equation.

3. x – 4 = 10 14

4. 15 = x + 1.1 13.9

–10 0 10

–90 0 90

Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequality and inverse operations.

Helpful Hint

Use an inverse operation to “undo” the operation in an inequality. If the inequality contains addition, use subtraction to undo the addition.

Using Addition and Subtraction to Solve Inequalities

Solve the inequality and graph the solutions.

x + 12 < 20 x + 12 < 20

–12 –12x + 0 < 8

x < 8

Since 12 is added to x, subtract 12 from both sides to undo the addition.

–10 –8 –6 –4 –2 0 2 4 6 8 10Draw an empty circle at 8.

Shade all numbers less than 8 and draw an arrow pointing to the left.

d – 5 > –7

Since 5 is subtracted from d, add 5 to both sides to undo the subtraction.

Draw an empty circle at –2.

Shade all numbers greater than –2 and draw an arrow pointing to the right.

+5 +5d + 0 > –2

d > –2

d – 5 > –7

Solve the inequality and graph the solutions.

–10 –8 –6 –4 –2 0 2 4 6 8 10

Solve the inequality and graph the solutions.

0.9 ≥ n – 0.3

Since 0.3 is subtracted from n, add 0.3 to both sides to undo the subtraction.

Draw a solid circle at 1.2.

Shade all numbers less than 1.2 and draw an arrow pointing to the left.

0 1 2

+0.3 +0.31.2 ≥ n – 0

1.2 ≥ n

0.9 ≥ n – 0.3

1.2

a. s + 1 ≤ 10

–1– 1 s + 0 ≤ 9

s ≤ 9

Since 1 is added to s, subtract 1 from both sides to undo the addition.

b. > –3 + t

Since –3 is added to t, add 3 to both sides to undo the addition.

Solve each inequality and graph the solutions.

s + 1 ≤ 10

> –3 + t

+3 +3

> 0 + t

t <

9

–10 –8 –6 –4 –2 0 2 4 6 8 10

–10 –8 –6 –4 –2 0 2 4 6 8 10

q – 3.5 < 7.5

+3.5 +3.5

q – 0 < 11

q < 11

Since 3.5 is subtracted from q, add 3.5 to both sides to undo the subtraction.

Solve the inequality and graph the solutions.

q – 3.5 < 7.5

–7 –5 –3 –1 1 3 5 7 9 11 13

Solve each inequality and graph the solutions.

1. 13 < x + 7

x > 6

2. –6 + h ≥ 15h ≥ 21

3. 6.7 + y ≤ –2.1

y ≤ –8.8

Multiplying or Dividing by a Positive Number

Solve the inequality and graph the solutions.

7x > –42

7x > –42

>

1x > –6

Since x is multiplied by 7, divide both sides by 7 to undo the multiplication.

x > –6

–10 –8 –6 –4 –2 0 2 4 6 8 10

3(2.4) ≤ 3

7.2 ≤ m (or m ≥ 7.2)

Since m is divided by 3, multiply both sides by 3 to undo the division.

0 2 4 6 8 10 12 14 16 18 20

Solve the inequality and graph the solutions.

r < 16

0 2 4 6 8 10 12 14 16 18 20

Since r is multiplied by ,

multiply both sides by the

reciprocal of .

Solve the inequality and graph the solutions.

Solve the inequality and graph the solutions.

4k > 24

k > 6

0 2 4 6 8 10 12 16 18 2014

Since k is multiplied by 4, divide both sides by 4.

–50 ≥ 5q

–10 ≥ q

Since q is multiplied by 5, divide both sides by 5.

Solve the inequality and graph the solutions.

5–5 0–10–15 15

g > 36

Since g is multiplied by ,

multiply both sides by the

reciprocal of .

36

25 30 3520 4015

Solve the inequality and graph the solutions.

If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality symbol to make the statement true.

Caution!

Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

Multiplying or Dividing by a Negative Number

Solve the inequality and graph the solutions.

–12x > 84

x < –7

Since x is multiplied by –12, divide both sides by –12. Change > to <.

–10 –8 –6 –4 –2 0 2 4 6–12–14

–7

Since x is divided by –3, multiply both sides by –3. Change to .

16 18 20 22 2410 14 26 28 3012

Solve the inequality and graph the solutions.

24 x (or x 24)

Solve each inequality and graph the solutions.

a. 10 ≥ –x

–1(10) ≤ –1(–x)

–10 ≤ x

Multiply both sides by –1 to make x positive. Change to .

b. 4.25 > –0.25h

–17 < h

Since h is multiplied by –0.25, divide both sides by –0.25. Change > to <.

–20 –16 –12 –8 –4 0 4 8 12 16 20

–17

–10 –8 –6 –4 –2 0 2 4 6 8 10

Solve each inequality and graph the solutions.

1. 8x < –24 x < –3 2. –5x ≥ 30 x ≤ –6

3. x > 20 4. x ≥ 6

5. A soccer coach plans to order more shirts for her team. Each shirt costs $9.85. She has $77 left in her uniform budget. What are the possible number of shirts she can buy?0, 1, 2, 3, 4, 5, 6, or 7 shirts

Warm UpSolve each equation. 1. 2x – 5 = –17

2.

Solve each inequality and graph the solutions.

4.

3. 5 < t + 9

–6

14

t > –4

a ≤ –8

Solving Multi-Step Inequalities

Solve the inequality and graph the solutions.

45 + 2b > 61

45 + 2b > 61–45 –45

2b > 16

b > 8

0 2 4 6 8 10 12 14 16 18 20

Since 45 is added to 2b, subtract 45 from both sides to undo the addition.

Since b is multiplied by 2, divide both sides by 2 to undo the multiplication.

8 – 3y ≥ 298 – 3y ≥ 29–8 –8

–3y ≥ 21

y ≤ –7

Since 8 is added to –3y, subtract 8 from both sides to undo the addition.

Since y is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤.

–10 –8 –6 –4 –2 0 2 4 6 8 10

–7

Solve the inequality and graph the solutions.

Solve the inequality and graph the solutions.

–12 ≥ 3x + 6–12 ≥ 3x + 6– 6 – 6

–18 ≥ 3x

–6 ≥ x

Since 6 is added to 3x, subtract 6 from both sides to undo the addition.

Since x is multiplied by 3, divide both sides by 3 to undo the multiplication.

–10 –8 –6 –4 –2 0 2 4 6 8 10

Solve the inequality and graph the solutions.

2 – (–10) > –4t

12 > –4t

–3 < t (or t > –3)

Combine like terms.Since t is multiplied by –4, divide

both sides by –4 to undo the multiplication. Change > to <.

–3

–10 –8 –6 –4 –2 0 2 4 6 8 10

Solve the inequality and graph the solutions.

–4(2 – x) ≤ 8

−4(2 – x) ≤ 8

−4(2) − 4(−x) ≤ 8 –8 + 4x ≤ 8

+8 +84x ≤ 16

x ≤ 4

Distribute –4 on the left side.

Since –8 is added to 4x, add 8 to both sides.

Since x is multiplied by 4, divide both sides by 4 to undo the multiplication.

–10 –8 –6 –4 –2 0 2 4 6 8 10

Solve the inequality and graph the solutions.

3 + 2(x + 4) > 3

3 + 2(x + 4) > 33 + 2x + 8 > 3

2x + 11 > 3– 11 – 11

2x > –8

x > –4

Distribute 2 on the left side.

Combine like terms.Since 11 is added to 2x, subtract

11 from both sides to undo the addition.

Since x is multiplied by 2, divide both sides by 2 to undo the multiplication.

–10 –8 –6 –4 –2 0 2 4 6 8 10

Solve each inequality and graph the solutions.

1. 13 – 2x ≥ 21 x ≤ –4

2. –11 + 2 < 3p p > –3

3. 23 < –2(3 – t) t > 7

4.

Example 1: x + 2 > 5

Example 2: 3x 6

Example 3: 6x + 12 -2x - 4

1. Describe the graphs of linear inequalities based on the given examples?

2. How does the graph of linear inequalities differ from linear equations?

3. What does the direction of the inequality symbol > or indicated?

< or ?

4. What are the solutions of the inequalities?

An _________ is an open sentence

using the relation symbols such as

>, < , and . _________ of

inequalities can be shown by

means of a graph.

INEQUALITY

SOLUTION

Solve and graph the solution

1. x – 4 > 2

2. x – 10 - 4

3. 2 + x 5

4. x + 8 < - 8

Solve and graph the solution.

1. y + 6 < 4

2. 6x - 18

3. 4x > 3x + 5

4. 3x < 2x + 6

1 whole. Show your solution. Box your final answer.P 269 a-d (4)P 271 1-10 (10)P274 a-b (2)P275 c (1)P277 Ind Prac 1-11 (11)

28 points (to be checked next year) MERRY CHRISTMAS!!!!