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Step-By-Step Guide to Solving Linear Equations & Inequalities

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Page 1: Tutorial linear equations and linear inequalities

Step-By-Step Guide to Solving Linear

Equations & Inequalities

Page 2: Tutorial linear equations and linear inequalities

Table of Contents

Open 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

Page 3: Tutorial linear equations and linear 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

Page 4: Tutorial linear equations and linear inequalities

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 0For example:

2x + 6 = 0

Page 5: Tutorial linear equations and linear inequalities

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.

Page 6: Tutorial linear equations and linear inequalities

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

Page 7: Tutorial linear equations and linear inequalities

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 -

Page 8: Tutorial linear equations and linear inequalities

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

Page 9: Tutorial linear equations and linear inequalities

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

Page 10: Tutorial linear equations and linear inequalities

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

Page 11: Tutorial linear equations and linear inequalities

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?

Page 12: Tutorial linear equations and linear inequalities

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

Page 13: Tutorial linear equations and linear inequalities

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

Page 14: Tutorial linear equations and linear inequalities

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

Page 15: Tutorial linear equations and linear inequalities

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

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?

Page 16: Tutorial linear equations and linear inequalities

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

Page 17: Tutorial linear equations and linear inequalities

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 =

36 6) Divide both sides by 9.

5) Simplify.

+2x +2x

- 19 -19

36x9 9 9

7) Simplify.

Can you check your answer? How?

Page 18: Tutorial linear equations and linear inequalities

)( 25211 xx

105211 xx

-5x

-5x 1026 x

-2

-2126 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

Page 19: Tutorial linear equations and linear inequalities

Applying equations in everyday life

To solve problems encountered in daily life, especially word problems, the following steps can help make finding the solution 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.

Page 20: Tutorial linear equations and linear inequalities

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?

J started with some money in their pocket. All J spent was $4.65 on lunch. J ended up with $7.39 in their pocket. Write an equation to find out how much money J started with.

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

Page 21: Tutorial linear equations and linear inequalities

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

The drop in temperature is 58 degrees Fahrenheit.

2. J 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, J started with $ 12,04

Page 22: Tutorial linear equations and linear inequalities

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?

Page 23: Tutorial linear equations and linear inequalities

1)1 batch 6 cans 3 ½ batch X cansWe can write the equations:

1 672

7.1 6.

23.7

21

x

x

x

x

George needs 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 XWe can write the equations :

1 3

18.1 3.18

54

xx

x

Amy’s mom brought

54 cookies

Page 24: Tutorial linear equations and linear inequalities

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?

Page 25: Tutorial linear equations and linear inequalities

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.

Page 26: Tutorial linear equations and linear inequalities

What’s a linear inequality in one variable?

A linear inequality is an open sentence which has the relation of ≥, >, ≤, or < and whose variable has an exponent of 1.

A linear 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

Page 27: Tutorial linear equations and linear inequalities

LESS THAN

GREATER THAN

LESS THAN OR EQUAL TO

GREATER THAN OR EQUAL TO

Symbols

Page 28: Tutorial linear equations and linear inequalities

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 < 5

Solve using addition:

53 x+3 +3

x < 8

Page 29: Tutorial linear equations and linear inequalities

Solving Using Subtraction

Subtract the same number from EACH side.

106 x

4x-6 -6

Page 30: Tutorial linear equations and linear inequalities

Solving using Multiplication

Multiply each side by the same positive number.

32

1x

6x

(2) (2)

Page 31: Tutorial linear equations and linear inequalities

Divide each side by the same positive number.

93 x

3x3 3

SOLVING USING DIVISION

Page 32: Tutorial linear equations and linear inequalities

Solving by multiplication of a negative #

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

Multiply by (-1).4 x

4x

(-1) (-1)

See the switch

Page 33: Tutorial linear equations and linear inequalities

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.

Page 34: Tutorial linear equations and linear inequalities

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

Page 35: Tutorial linear equations and linear inequalities

Solutions continued…

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

All real numbers greater than -2

x > -2

Page 36: Tutorial linear equations and linear inequalities

Solutions continued….

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

All real numbers less than or equal to 1

1x

Page 37: Tutorial linear equations and linear inequalities

Solutions continued…

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

All real numbers greater than or equal to -3

3x

Page 38: Tutorial linear equations and linear inequalities

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.

Page 39: Tutorial linear equations and linear inequalities

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.

Page 40: Tutorial linear equations and linear inequalities

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!

Page 41: Tutorial linear equations and linear inequalities

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