Chapter 3. linear equation and linear equalities in one variables
1.
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. 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
5. 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.
6. 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
x 3 6 (opposite) of subtracting 3 is _________________ 3.
x 3 3 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
x 9 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. The process for solving equations using subtraction is very
similar to solving equations using addition.
How do you undo addition?
Example: First, we must “undo” the plus 2. The inverse
x 2 5 (opposite) of adding 2 is ______________ 2.
x 2 -2 5 -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
x 3 equation.
Ex1) x 7 9 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 What does this mean?
x 12
x = -12 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
-x = -5 x 2 7 z 6 ( x) 7
3 5
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. • 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. 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 2 , you will need to use
multiplication. 4
x In this problem, the x is being divided by 4. To
2 solve for x, we will need to do the inverse of
4 dividing by 4, which is multiplying by 4.
x **Don’t forget that you will need to do this to
4 42 BOTH sides of the equation to keep it balanced!
4
Ex) a a
x 8 10 10
2 2
14. To solve equations like 4x 32 , we can use division.
4x 32 In this problem, the x is being multiplied by 4.
To solve for x, we will need to do the inverse of
4x 32 multiplying by 4, which is DIVIDING by 4.
**Don’t forget that you will need to do this to
4 4 BOTH sides of the equation to keep it balanced!
x 8
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 1to 7
which require more than one step 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 7 There is nothing to simplify What goes
first?
1 1 Simplify to get a new equation.
3x 6 What goes next?
3 3 How can you check you answer?
Do I expect you to show work on
x 2 assignments, quizzes and tests?
16. Lots of steps! Where should you start??
3( x 2) 4 x 12 1
distribute 3( x 2) 4 x 12 1 simplify
Combine like terms 3x 6 4x 13
7x 6 13
6 6
7x 7
7 7
x 1
17. 1st: Variables to one side How do you decide who to
move?
2nd: Constants to the other side Who must you move?
1) Which side has the smaller
7x + 19 = -2x + 55 coefficient?
+2x +2x 2) Add 2x to both sides.
3) Simplify.
9x + 19 = 55
- 19 -19 4) Subtract 19 from both sides.
9x = 36 5) Simplify.
9x 36 6) Divide both sides by 9.
9 9 7) Simplify.
x 4 Can you check your answer? How?
18. The Graph of Solutions of Equations
11x 2 5(x 2)
11x 2 5x 10
-5x -5x
6x 2 10 6x 12
-2 -2 6 6
6x 12 x 2
The graph is
-5 -4 -3 -2 -1 0 1 2 3 4 5
19. 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.
20. 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.
Can you solve one – step problems involving addition and subtr
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 x 4
54 4 x
58 x
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
x 4, 65 7,39
x 7,39 4, 65
x 12, 04
So, I started with $ 12,04
22. 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?
23. 1) 1 batch 6 cans
2) The number of team members :
3 ½ batch X cans
15+2+1=18 persons
We can write the equations:
1 person 3 cookies
18 persons X
1 6
7 x We can write the equations :
2 1 3
7
x.1 6. 18 x
2
x 3.7 x.1 3.18
x 21 x 54
So, George need 21 cans of
chili to make 3 and a half So, Amy’s mom bring 54 cookies
batches.
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
400 125 50 x
275 50 x
275
x
50
5,5 x So, the number of hours of labor = 5,5 hours.
26. 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: ax b...0 ,
where ... can be one of , , ,or
27.
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:
x–3<5
Add the same number to EACH side.
x 3 5
+3 +3
x<8
32. Solving by multiplication of a
negative #
Multiply each side by the same negative number and REVERSE the
inequality symbol.
Multiply by (-1).
(-1) x 4 (-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
34. 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.
35. 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
36. Solutions continued…
-5 -4 -3 -2 -1 0 1 2 3 4 5
All real numbers greater than -2
x > -2
38. Solutions continued…
-5 -4 -3 -2 -1 0 1 2 3 4 5
All real numbers greater than or equal to -3
x 3
39. 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
x 1
-5 -4 -3 -2 -1 0 1 2 3 4 5
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.
40. 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.
41. 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!
42. • Answer:
Width = x cm, length = (x+6) cm.
Perimeter = 2p + 2l.
2p 2l 40
2( x 6) 2x 40 x cm
2x 12 2x 40
4x 12 40 (x+6) cm
4x 40 12
4x 28
4 x 28
Since a length and a width cannot be negative, then its
4 4 solutions is
x 7 0 x 7
