Solving Inequalities > < ≥ ≤

Inequalities H.notebook September 27, 2016

What Does the Prefix "In" mean when added to a word?

Solving InequalitiesAn inequality is a number sentence much like an equation, but rather than an equal sign in the middle, there is...

> < ≥ ≤

The solution to an equation looks like...

x = 3

The solution to an inequality looks like

x < 5

In both cases, the solution makes the number sentence a true statement.


Solutions to Inequalities

The number of cars in the parking lot is greater than 5.

c > 5

How many possible numbers for c make a true statement?

Is the number a solution of

2x + 1 > -3

A. -3 B. -1

< > ≤ ≥

less than

below

fewer than

greater than

more than

above

Exceeds

less/equal

at most

no more than

up to and including

greater/equal

at least

no less than

at minimum

The symbols...

and the words which imply them.


Practice writing the inequalities

1. There is more than 21 pilots in the airport.

2. I ate at most 43 cookies in one sitting.

3. The fire was burning on all floors below the 7th.

4. She will receive no less than an 81 on her test.

5. There were fewer than 4 officers at the scene.

Graphing Inequalities on a Number Line

Inequality Symbol

<

>

≥

≤

I ran no more than 13 miles.

I walked less than 2 miles.


Write the inequality represented by the graph.

-3

4

27

-11

Practice problemsAre the numbers solutions to the inequalities?

1. 4x + 1 < 13 5 0 3

2. -3x + 5 ≥ 2 -2 2 1

Write the inequality and graph.

3. I watched no less than 10 episodes.

4. Berkeley ran the ball fewer than 24 times.

5. We traveled at most 900 miles.


Grab a Calculator

Take out Your Notebook

Bell RingerWrite the inequality to represent the situation and graph on a

number line.

There are no more than 11 characters in the movie.

Solving Inequalities

So remember that whole "solving equations" thing we did? Well it hasn't gone anywhere, and it's only coming back!

We solve inequalities the same way as we solved

equations, using _____________ _____________ .

x + 4 > -8

Check your

answer!!!


1. x - 5 ≤ 19

2. x -10 > -1

3. x + 9 < 6

One Rule to Note!!!

When multiplying or dividing both sides of inequality by a NEGATIVE number, you MUST FLIP the inequality symbol.

-4x ≥ 8


1. -5x > 30

2. 4x + 2 < 22

3. -7x ≥ 42

4. 9 - x ≤ 11

The graph below represents which inequality?

-6

A. x > -6

B. x < -6

C. x < -6

D. x > -6


6 - 4x > -30

A.x > -9 B.x > 9 C.x < -9 D.x < 9

Solve Multi-Step Inequalities

Once again, we will solve multi-step inequalities just as we solved multi-step equations.

4x + 3 > -x + 13

Solve and graph the solutions to the inequalities.

1. -2x - 8 < -5x + 25

2. 3(x - 4) ≥ 7x + 8


Grab a Calculator


Answer the inequality below in your notebook...

-5x > 40

A.x > 8 B.x > -8 C.x < 8 D.x < -8

Bell Ringer

Solve the Inequalities

1. 6(x + 3) > -36

2. 4x - 8 > 6x + 24


Solve the Multi-Step Equations and sketch the solutions

1. 4x - 3 < 2x + 21

2. -6x + 2 > x + 16

3. -x + 10 > - 5x +2

Name _____________

Word Problems: Defining a Variable

When turning a word problem into an equation/inequality we need to determine what the variable

will be.

Tommy has $40 in his wallet. He will spend two dollars on burgers. He has at least eight dollars left. What is the maximum amount of burgers he could have bought?

The variable will be the object(s) asked for in the question.


Rate Constant Total

Word Problems Featuring InequalitiesYour cell plan costs $40.00 a month plus $.14 for each text message you send or receive. You have at most $45 to spend on your cell phone bill.

1. Write an equation which represents this scenario.

2. What is the maximum number of texts you can send.

3. Explain your answer to number 2.


A used car is going to cost Chasity at least $555. She has managed to save up $222. If her job pays her $15 an hour, how many hours must she work in order to be able to afford the car?

1. Write an equation which represents this scenario.

2. What is the minimum number of whole number hours she can work?

A. h ≥ 22 B. h ≤ 22 C. h ≥ 23 D. h ≤ 23

Grab a Calculator


3x + 2 > 4x + 8

A.x > -6 B.x > 6 C.x < -6 D.x < 6

Bell Ringer


Grab a Calculator



-5x > 40

A.x > 8 B.x > -8 C.x < 8 D.x < -8

Bell Ringer

A compound inequality consists of two distinct inequalities joined by the word and or the word or.

Compound Inequalities

And Inequality

4

9

4 9

x > 4

x < 9

x > 4 and x < 9

Or Inequality

10

2

2 10

x > 10

x < 2

x < 2 or x > 10

4 < x < 9


Graph the inequalities

1. x > 3 and x < 11

2. x < -1 or x > 4

3. 7 < x < 12

Solving Compound "And" Inequalities

3 < m + 8 < 12


-3 < x-4 < -1

-8 < 2y-6 < -2


Solving Compound "Or" Inequalities

4x > 8x + 5 < 1 or

3x - 1 < 11 2x > 10or


To achieve a B in your college English class, you must achieve a test average between 84 and 86. You scored 86, 85, and 80 on the first three exams. What possible scores can you earn on the fourth and final to earn a B.

What are the solutions of

3x + 2 < -7 OR -4x + 5 < 1


Solve and Graph

5x + 1 < 11 AND 6x > 30

Get a Calculator

Bell Ringer

6 < 3x + 9 ≤ 15Solve and Graph the Results


Interval Notation

When writing the solution set to an inequality, we may choose to use, (or be asked to use), interval notation. An interval is a portion of the number line.

Parenthesis - Use parentheses when < or > are used to describe an interval's endpoint.

Brackets - Use brackets when < or > are used to describe an interval's endpoints.

Infinty - Use ∞ when the interval continues off in a positive direction. Use -∞ when the interval continues off in a negative direction.

Solve, graph, and use interval notation to describe the solution.

1. 3 < 4x - 5 < 15

2. 5x - 3 > 7 OR 4x - 6 < -10

3. -9 < 3m + 6 < 18


Domain and Range

Domain

Range

Domain

Range


Domain

Range

Write the inequality which represents the word sentence.1. There are at least 20 kittens in the box.

2. I’m going to eat less than 20 of those kittens.

3. The number of kittens I ate exceeds 13.

Graph the inequalities.4. x < 8 5. x ≥ ‐1

6. 6 ≤ x

Solve the inequalities. Graph their solutions.

7. x + 5 > 8 8. ‐5x < 20 9.≤ 6

10. 4x – 8 < 8x + 12 11. 6x – 4 ≥ 812. ‐29 < 9x ‐ 2


2 < 2x + 6 < 16Solve

Graph

Interval Notation

Shane wants to get into Harvard. He needs more than a 91 average in History to get into their history program. So far, his test grades are 89, 93, and 85. He has one more test to take. Write and solve the inequality to determine what he must score in order to meet his goal.

Bell Ringer


Absolute Value

The absolute value of a number is the distance the number is from zero.

4-6 0

A "distance" can't be negative.

-6

4

= 6

= 4

Solving Absolute Value Equations

x + 2= 9

0

x + 2= 9


An absolute value

problem will

always have

Two Answers

x - 2 = 9

Note: Make sure the abs val is by itself before you split up the problem.


2x + 4 - 5 = 7

-3 4x = -36


x + 4 - 8 = 22

x + 2 = -6


Practice Problems

x - 5 = 4

x - 3 = 10

2x + 6 = 12

x + 6 - 12 = 93

Documents

Solving Inequalities > < ≥ ≤