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Tutorial linear equations and linear inequalities
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
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 :


x 7 15

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

x 8
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
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
To solve equations like x 3 6 , you can use mental
math.
Ask yourself, “What number minus 3 is equivalent to 6?”

x

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

In this strategy, we balance the sides of the equation as we
solve for the variable.
First, we must “undo” the minus 3. The inverse
(opposite) of subtracting 3 is _________________ 3.

x 3 6
x

3

3

x 9

6

3

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.
• 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
The process for solving equations using subtraction is very
similar to solving equations using addition.

How do you undo addition?

x 2 5

Example:

5 -2

x 2 -2

x 3
Ex1) x 7

9

First, we must “undo” the plus 2. The inverse
(opposite) of adding 2 is ______________ 2.
But 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.

Ex2) x 15

10
In some problems, you will see a negative sign in front
of the variable you are solving for.
Example:

x 12
x = -12

x 12

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.
4
2
6 ( x)
z
-x = -5
x 2 7
3
5

7
• 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
Objectives: To solve linear equations using multiplication and division and to
use linear equations to solve word problems involving real-world situations

To solve equations like x
4
multiplication.

x
2
4
x
4
42
4

x 8

2 , you will need to use

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!

Ex)

a
2

10

a
10
2
Solving Equations Using Division
To solve equations like 4x

4x 32

4x
4

32
4

x 8

32

, we can use division.

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!
Objectives: To use 2 or more steps to solve a linear equation and to use
multi-step equations to solve word problems

You will be asked to solve problems like 3x 1 7
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 subtract
Then  Undo multiplication and division

3x 1
1
3x
3

7
1
6
3

x

2

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

What goes next?

How can you check you answer?
Do I expect you to show work on
assignments, quizzes and tests?
Lots of steps! Where should you start??

3( x 2) 4 x 12 1
3( x 2) 4 x 12 1
Combine like terms
3x 6 4x 13
distribute

7x 6 13
6
6
7x 7
7 7

x 1

simplify
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
+2x
+2x
9x + 19 = 55
- 19 -19
9x
= 36
9x 36
9
9

x 4

1) Which side has the smaller
coefficient?
2) Add 2x to both sides.
3) Simplify.

4) Subtract 19 from both sides.
5) Simplify.
6) Divide both sides by 9.
7) Simplify.
Can you check your answer? How?
The Graph of Solutions of Equations

11x

5(x

2

11x
-5x

2

6x

2
-2

2)

5x 10
-5x

10
-2

12
6

12

6x

6x
6

x

2

The graph is
-5
4

-4
5

-3

-2

-1

0

1

2

3
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.
Can you solve one – step problems involving addition and subtr
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.
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

54
54

x
4

58

4

x

x

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

x 4, 65 7,39
x

7,39 4, 65

x 12, 04

So, J started with $ 12,04
Ex 1.) It takes 6 cans of chili to make one batch of George’s extraspecial 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
7
2

6
x

7
x.1 6.
2
x 3.7
x

21

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  X
We can write the equations :

1 3
18 x
x.1 3.18
x 54
Amy’s mom brought
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 x

So, the number of hours of
labor = 5.5 hours.

400 125 50 x
275 50 x
275
x
50
5,5 x
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: ax b...0
where ... can be one of

, , ,or
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.

Solve using addition: – 3 < 5
x
Add the same number to EACH side.

x 3 5
+3

x<8

+3
Solving Using Subtraction

Subtract the same number from EACH side.

x 6 10
-6

-6

x 4
Solving using Multiplication

Multiply each side by the same positive number.

1
(2) x
2

3 (2)

x 6
Divide each side by the same positive number.

3x 9
3

3

x 3
Solving by multiplication of a negative #
Multiply each side by the same negative number and REVERSE the
inequality symbol.

(-1)

x

4

(-1)

Multiply by (-1).

See the switch

x

4
Solving by dividing by a negative #
Divide each side by the same negative number and reverse
the inequality symbol.

2x

6

-2

-2

x

3

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.
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

All real numbers greater than -2

x > -2

4

5
Solutions continued….

-5
4

-4

-3

-2

-1

0

1

5

All real numbers less than or equal to 1

x 1

2

3
Solutions continued…

-5 -4

-3

-2

-1

0

1

2

3

All real numbers greater than or equal to -3

x

3

4

5
Did you notice,
Some of the dots were
solid and some were open?

x 2
-5 -4 -3 -2 -1

0

1

2

3

4

5

-5 -4 -3 -2 -1

0

1

2

3

4

5

x 1
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
point too.

or

then the dot is solid, because it can be that
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.


2p

2l
2( x

40
6)

2x

2x

12

2x

4x

12

40 12

4x

28

x cm

40

4x

40

4 x 28
4
4
x 7

40
(x+6) cm

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

0

x 7

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Tutorial linear equations and linear inequalities

  • 2. 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
  • 3. 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 :  x 7 15 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 x 8
  • 4. 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
  • 5. 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.
  • 6. 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
  • 7. To solve equations like x 3 6 , you can use mental math. Ask yourself, “What number minus 3 is equivalent to 6?” x This strategy can work for easier problems, but we need a better plan so we can solve more difficult problems. In this strategy, we balance the sides of the equation as we solve for the variable. First, we must “undo” the minus 3. The inverse (opposite) of subtracting 3 is _________________ 3. x 3 6 x 3 3 x 9 6 3 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.
  • 8. • 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
  • 9. Solving Linear Equations Using Subtraction The process for solving equations using subtraction is very similar to solving equations using addition. How do you undo addition? x 2 5 Example: 5 -2 x 2 -2 x 3 Ex1) x 7 9 First, we must “undo” the plus 2. The inverse (opposite) of adding 2 is ______________ 2. But 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. Ex2) x 15 10
  • 10. In some problems, you will see a negative sign in front of the variable you are solving for. Example: x 12 x = -12 x 12 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. 4 2 6 ( x) z -x = -5 x 2 7 3 5 7
  • 11. • 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?
  • 12. 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
  • 13. Solving Equations Using Multiplication Objectives: To solve linear equations using multiplication and division and to use linear equations to solve word problems involving real-world situations To solve equations like x 4 multiplication. x 2 4 x 4 42 4 x 8 2 , you will need to use 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! Ex) a 2 10 a 10 2
  • 14. Solving Equations Using Division To solve equations like 4x 4x 32 4x 4 32 4 x 8 32 , we can use division. 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!
  • 15. Objectives: To use 2 or more steps to solve a linear equation and to use multi-step equations to solve word problems You will be asked to solve problems like 3x 1 7 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 subtract Then  Undo multiplication and division 3x 1 1 3x 3 7 1 6 3 x 2 There is nothing to simplify  What goes first? Simplify to get a new equation. What goes next? How can you check you answer? Do I expect you to show work on assignments, quizzes and tests?
  • 16. Lots of steps! Where should you start?? 3( x 2) 4 x 12 1 3( x 2) 4 x 12 1 Combine like terms 3x 6 4x 13 distribute 7x 6 13 6 6 7x 7 7 7 x 1 simplify
  • 17. 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 +2x +2x 9x + 19 = 55 - 19 -19 9x = 36 9x 36 9 9 x 4 1) Which side has the smaller coefficient? 2) Add 2x to both sides. 3) Simplify. 4) Subtract 19 from both sides. 5) Simplify. 6) Divide both sides by 9. 7) Simplify. Can you check your answer? How?
  • 18. The Graph of Solutions of Equations 11x 5(x 2 11x -5x 2 6x 2 -2 2) 5x 10 -5x 10 -2 12 6 12 6x 6x 6 x 2 The graph is -5 4 -4 5 -3 -2 -1 0 1 2 3
  • 19. 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.
  • 20. Can you solve one – step problems involving addition and subtr 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.
  • 21. 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 54 54 x 4 58 4 x x 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 x 4, 65 7,39 x 7,39 4, 65 x 12, 04 So, J started with $ 12,04
  • 22. Ex 1.) It takes 6 cans of chili to make one batch of George’s extraspecial 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?
  • 23. 1) 1 batch  6 cans 3 ½ batch  X cans We can write the equations: 1 7 2 6 x 7 x.1 6. 2 x 3.7 x 21 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  X We can write the equations : 1 3 18 x x.1 3.18 x 54 Amy’s mom brought 54 cookies
  • 24. 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?
  • 25. 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 x So, the number of hours of labor = 5.5 hours. 400 125 50 x 275 50 x 275 x 50 5,5 x
  • 26. 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: ax b...0 where ... can be one of , , ,or
  • 28. 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. Solve using addition: – 3 < 5 x Add the same number to EACH side. x 3 5 +3 x<8 +3
  • 29. Solving Using Subtraction Subtract the same number from EACH side. x 6 10 -6 -6 x 4
  • 30. Solving using Multiplication Multiply each side by the same positive number. 1 (2) x 2 3 (2) x 6
  • 31. Divide each side by the same positive number. 3x 9 3 3 x 3
  • 32. Solving by multiplication of a negative # Multiply each side by the same negative number and REVERSE the inequality symbol. (-1) x 4 (-1) Multiply by (-1). See the switch x 4
  • 33. Solving by dividing by a negative # Divide each side by the same negative number and reverse the inequality symbol. 2x 6 -2 -2 x 3 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.
  • 34. 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
  • 35. Solutions continued… -5 -4 -3 -2 -1 0 1 2 3 All real numbers greater than -2 x > -2 4 5
  • 36. Solutions continued…. -5 4 -4 -3 -2 -1 0 1 5 All real numbers less than or equal to 1 x 1 2 3
  • 37. Solutions continued… -5 -4 -3 -2 -1 0 1 2 3 All real numbers greater than or equal to -3 x 3 4 5
  • 38. Did you notice, Some of the dots were solid and some were open? x 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 1 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 point too. or then the dot is solid, because it can be that
  • 39. 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.
  • 40. 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!
  • 41. Answer: Width = x cm, length = (x+6) cm. Perimeter = 2p + 2l.  2p 2l 2( x 40 6) 2x 2x 12 2x 4x 12 40 12 4x 28 x cm 40 4x 40 4 x 28 4 4 x 7 40 (x+6) cm Since a length and a width cannot be negative, then its solutions is 0 x 7