LINEAR EQUATIONSAND

LINEAR INEQUALITIESIN ONE VARIABLE

Maping conceptOpen Sentences

Understanding Linear Equations in One Variable

Solving Linear Equations in One Variable

Applying Equations in Everyday life

The Graph of Solutions of Equations

Understanding Inequalities

Solving Linear Inequalities in One Variable

The Graph of an Inequality

Applying Inequalities

Open Sentences A sentence which contains one or more variables such that its

truth (being true or false) can not be determined is called an open sentence.

Example :

A variable is a symbol that can be replaced with an arbitrary number.

From the example, X is a variable.

The subtitute for a variable which makes an open sentence a true sentence is called a solution.

From the example, if x is replaced with 8, the sentence will be true.

Thus, its solution is

7 15x

8x

What’s a Linear Equation in One

Variable ?

A linear equation in one variable is an equation that can be written in the form

ax + b = 0

Where a 0 For example:

2x + 6 = 0

Solving Equations

To solve an equation means to find all values that make the equation a true statement. Such values are called solutions, and the set of all solutions is called the solution set.

Solving Equations by Using Addition and Subtraction

Objectives:• * To solve linear equations using addition

and subtraction• * To use linear equations to solve word

problems involving real-world situations

• Your goal is to isolate the variable and keep sides balanced

• Watch integer signs and remember the rules• Fraction and decimal rules still apply

Solving Linear Equations Using Addition

In this strategy, we balance the sides of the equation as we solve for the variable.

To solve equations like , you can use mental math.Ask yourself, “What number minus 3 is equivalent to 6?”

63 x

This strategy can work for easier problems, but we need a better plan so we can solve more difficult problems.

First, we must “undo” the minus 3. The inverse (opposite) of subtracting 3 is _________________ 3.

63 x

But we have to be fair! If we are going to add 3 to one side of the equation, we MUST add 3 to the other side to BALANCE the equation.9x

x

3+ 3+6=3x -

Subtraction Property of Equality

• RULE: Subtract the same amount from each side of the equation to keep the equation balanced

• Algebraic symbols: for any numbers a, b, and c, if a=b, then a-c= b-c

Solving Linear Equations Using Subtraction

52x

First, we must “undo” the plus 2. The inverse (opposite) of adding 2 is ______________ 2.52 xBut we have to be fair! If we are going to subtract 2 from one side of the equation, we MUST subtract 2 from the other side to BALANCE the equation.3x

The process for solving equations using subtraction is very similar to solving equations using addition.How do you undo addition?

Example:

- 2 - 2

97x)1Ex 1015x)2Ex

Negative Signs with VariablesIn some problems, you will see a negative sign in front of the variable you are solving for.

Example: 12 x What does this mean?

The negative sign means “the opposite of x,” meaning the opposite sign, positive or negative.

To solve, we change the sign of the other side of the equation.

12x

-x = -5

x = -12

72 x52

z34

7)(6 x

Solving Equations by Using Multiplication and Division

• Focus: How can equations be used to find how long it takes light to reach Earth?

• What is the unknown quantity in the equation given in the example?

• What variable represents the unknown quantity in the equation?

• What do you need to accomplish to solve this equation?

To solve multiplication and division equations

• REMEMBER: Your goal is to isolate the variable!

• Multiplication Property: Multiply each side of the equation by the same number to keep the equation balanced

• Division Property: Divide both sides of the equation by the same number to keep the equation balanced

Solving Equations Using Multiplication

To solve equations like , you will need to use multiplication.

24

x

24

x In this problem, the x is being divided by 4.

To solve for x, we will need to do the inverse of dividing by 4, which is multiplying by 4. **Don’t forget that you will need to do this to BOTH sides of the equation to keep it balanced!

244

4

x

8x Ex)10

2

a10

2

a

Objectives: To solve linear equations using multiplication and division and to use linear equations to solve word problems involving real-world situations

Solving Equations Using Division

To solve equations like , we can use division.324 x

324 x In this problem, the x is being multiplied by 4. To solve for x, we will need to do the inverse of multiplying by 4, which is DIVIDING by 4. **Don’t forget that you will need to do this to BOTH sides of the equation to keep it balanced!

4

32

4

4x

8x

You will be asked to solve problems like which require more than one step to

solve.Guidelines for Solving Multi-Step Equations: 1) Simplify both sides if necessary distribute & combine like terms 2) To solve you must undo the order of operations BACKWARDS!

Start with Undo add or subtractThen Undo multiplication and division 13463 xx

713 x

713 x1 1

63 x3 3

2x

There is nothing to simplify What goes first?Simplify to get a new equation.

What goes next?

Multi-Step EquationsObjectives: To use 2 or more steps to solve a linear equation and to use multi-step equations to solve word problems

How can you check you answer?Do I expect you to show work on assignments, quizzes and tests?

1124)2(3 xx13463 xx

1367 x6 6

77 x7 7

1x

simplifydistribute

Combine like terms

Lots of steps! Where should you start?? 1124)2(3 xx

1st: Variables to one side How do you decide who to move?2nd: Constants to the other side Who must you move?

7x + 19 = -2x + 55

4x

1) Which side has the smaller coefficient?

2) Add 2x to both sides.3) Simplify.9x + 19 = 554) Subtract 19 from both sides.9x = 366) Divide both sides by 9.

5) Simplify.

+2x +2x

- 19 -19

36x9 9 9 7) Simplify.

Can you check your answer? How?

)( 25211 xx

105211 xx

-5x-5x

1026 x

-2 -2

126 x

6

12

6

6

x

2x

-5 -4 -3 -2 -1 0 1 2 3 4 5

The graph is

The Graph of Solutions of Equations

Applying equations in everyday life

To solve problems encountered in daily life, especially the word problems, the following steps can help make the solutin finding easier.– If you feel the need for a diagram or a sketch, e.g.,

for a problem related to geometry, make a diagram or a sketch based on the story.

– Translating the story into a mathematical sentence or mathematical model in the form the equations.

– Solve the equations.

Several record temperature changes have taken place in Spearfish, South Dakota. On January 22, 1943, the temperature in Spearfish fell

from 54 degrees Fahrenheit at 9:00 am to -4 degrees Fahrenheit at 9:27 am. By how many degrees did the temperature fall?

You started with some money in your pocket. All you spent was $4.65 on lunch. You ended up with $7.39 in your pocket. Write an

equation to find out how much money you started with.

Problems

Can you solve one – step problems involving addition and subtraction?

54 4

54 4

58

x

x

x

1. First temperature = 54 degrees Fahrenheit, Last temperature = (-4) degrees Fahrenheit, Let x = the fall of temperature. Write an equations to solve the problem, the equations is

So, the fall of temperature is 58 degrees Fahrenheit.

2. Let I started with X money, spent : $ 4,65 and end up : $ 7,39. Then, write an equations to find out the value of X, the equations is

4,65 7,39

7,39 4,65

12,04

x

x

x

So, I started with $ 12,04

Ex 1.) It takes 6 cans of chili to make one batch of George’s extra-special chili-cheese dip. How many cans does it take to make 3 and a half batches? Write an equation and solve it. Be sure to state what your variables represent.

Ex 2.) Amy’s mom has been passing out cookies to the members of Amy’s soccer team. If each of the team members (including 15 players, 2 coaches, and a manager) receive 3 cookies with no extra cookies left, how many cookies did Amy’s mom bring? Write an equation an solve it.

Can you solve equations involving multiplication and division?

1) 1 batch 6 cans 3 ½ batch X cans

We can write the equations:

1 672

7.1 6.

23.7

21

x

x

x

x

So, George need 21 cans of chili to make 3 and a half batches.

2) The number of team members : 15+2+1=18 persons

1 person 3 cookies 18 persons X

We can write the equations :

1 3

18.1 3.18

54

xx

x

So, Amy’s mom bring 54 cookies

Ex) The bill (parts and labor) for the repair of a car was $400. The cost of parts was $125, and the labor cost was $50 per hour. Write and solve and equation to find the number of hours of labor.

Can you solve multi-step equations? How do you know what operation to undo first?

The bill (parts and labor) = $ 400.Cost of parts = $ 125.Cost of labor per hour = $ 50.Let X is the number of hours of labor.The equation is

400 125 50

400 125 50

275 50

275

505,5

x

x

x

x

x

So, the number of hours of labor = 5,5 hours.

What’s a linier inequality in one variable?

A linier inequality is an open sentence which has the relation of ≥, >, ≤, or < and whose

variable has an exponent of 1.

A linier inequality which has one variable is called a linear inequality in one variable (LIOV).

The general form of a LIOV: , where ... can be one of

, , ,or ...0ax b

LESS THAN

GREATER THAN

LESS THAN OR EQUAL TO

GREATER THAN OR EQUAL TO

Symbols

Solving an Inequality

Solving a linear inequality in one variable is much like solving a linear equation in one variable. Isolate the variable on one side using inverse operations.

Add the same number to EACH side.

x – 3 < 5Solve using addition:

53 x+3 +3

x < 8

Solving Using Subtraction

Subtract the same number from EACH side.

106 x

4x-6 -6

Solving using Multiplication

Multiply each side by the same positive number.

32

1x

6x

(2) (2)

Divide each side by the same positive number.

93 x

3x3 3

SOLVING USING DIVISION

Solving by multiplication of a negative #

Multiply each side by the same negative number and REVERSE the inequality symbol.

4 x Multiply by (-1).

4x

(-1) (-1)

See the switch

Solving by dividing by a negative #

Divide each side by the same negative number and reverse the inequality symbol.

62 x

3x

-2 -2

When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement.

THE TRAPH

Solutions….

You can have a range of answers……

-5 -4 -3 -2 -1 0 1 2 3 4 5

All real numbers less than 2

X < 2

Solutions continued…

-5 -4 -3 -2 -1 0 1 2 3 4 5

All real numbers greater than -2

x > -2

Solutions continued….

-5 -4 -3 -2 -1 0 1 2 3 4 5

All real numbers less than or equal to 1

1x

Solutions continued…

-5 -4 -3 -2 -1 0 1 2 3 4 5

All real numbers greater than or equal to -3

3x

Did you notice,Some of the dots were solid and some were open?

-5 -4 -3 -2 -1 0 1 2 3 4 5

2x

-5 -4 -3 -2 -1 0 1 2 3 4 5

1x

Why do you think that is?If the symbol is > or < then dot is open because it can not be equal.

If the symbol is or then the dot is solid, because it can be that point too.

Applying inequlities in everyday life

To solve inequality problems in the form of word problems, translate the problems first into inequalities, and then solve them. If necessary, make a diagram or sketch to make the solution finding easier.

Problems

• The length of a rectangle is 6 cm longer than its width and its perimeter is less than 40 cm. If the rectangle’s width is x, then form an inequality in x and solve it!

• Answer:Width = x cm, length = (x+6) cm.Perimeter = 2p + 2l.

2 2 40

2( 6) 2 40

2 12 2 40

4 12 40

4 40 12

4 28

4 28

4 47

p l

x x

x x

x

x

x

x

x

Since a length and a width cannot be negative, then its solutions is 0 7x

(x+6) cm

x cm