buksis-4.4 revisi akhir
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at are you goingto learn? To d efine ineq ua lity
To d efine a linea r
inequality w ith one
variable
Key Terms:
inequality
linea r Ineq ua lity w ith
one va riable
solution and
solution set
Definition of Linear Inequality withOne Variable
Consider the number of students in your class. How
many students are in your class?
If the sentence The number of students in this
class is less than 25 persons is grouped according
to the phrases, we get
Let n be the number of students in this
class, then n < 25.
Now look at Figure 4.14 below.
Englishexpression
Number ofstudents inthis class
Is less than 25
44..44
MathematicalExpression
Max60
Km i
(iii)
(ii)
(iv)
17 years
Maximum Passengers6 people
Passengers cannot exceed
15 eo leFigure 4.14
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Look at Figure 4.14 that describes a real life situation.
i) it means that maximum speed is 60 km/hr.
ii) it means that the person to watch the movie must be 17
years old or more.
iii) it means that the number of passengers of the car cannot
be more than 6.
iv) it means that the maximum number of the
passengers is 15.
Work in groups or pairs.Using Figure 4.14. Answer the following questions1. Express your opinion, why is there a rule in each figure above?2. Let t be the speed of a car, m be the age of visitors, s be the numberof passengers of the car, h be the number of passengers of a ship.Write the condition of t, m, s and h in the mathematical expression.
Look again at your answer to Problem 2.Problem 3
a). Does each requirement that you have written has a
variable?
b). How many variables are there in each requirement?
c). What is the power of the variable?
d). Which notation do you use in your answer to Problem 2?
( =, , , )
e). In your answers to problem 2, which ones are open
sentences?
An open sentence using the sign >, ,
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From your answer to problem 2, which sentence is called a
linear inequality with one variable?
Figure 4.14 gives some examples of a real life situation
related to the linear inequality with one variable. Find
another example of daily life situation related to weight,
height, square, volume, report grades or others which can
be stated in a linear inequality with one variable.
Ida has 5 packs of writing books. Diah has 3 packs ofwriting books. The number of writing books in each pack is
the same. Ida gives 3 books to Susi and Diah receives extra 9
books from her mother.
Problem 4
The number of Diahs books is more than the number of
Idas books. If each pack contains n pieces of books,
a). write the relation between 5n 3 and 3n + 9.
b). find the value of n so that it holds for that relation.
c). find the value of n so that it does not hold for that
relation.
Every inequality contains variables. The substitution of a variable
that makes the sentence true is called a solution of the inequality.
The set of all solutions is called a solution set of that inequality.
-5 is a solution of the inequality 2x 5 < -x + 2, because2.(-5) 5 < -(-5) + 2 is a true statement. 4 is not a solution
of the inequality 4t 12 > 2t + 1, because 4.(4) 12>2.(4) + 1 is a false statement.
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Solving A Linear Inequality with One
Variable
Sketching the graph of solution in a line number
Look at the following line number and then answer the
questions below.
1
02
-1-3
-4
-2
-5 4
3
5
What numbers are solutions of the inequality x < 3?
Is 4 a solution of that inequality?
Is 3 a solution of that inequality?
Is 2 a solution of that inequality?
Is 1 a solution of that inequality?
Is 0 a solution of that inequality?
Is -1 a solution of that inequality?
Is -2 a solution of that inequality?
Is -3 a solution of that inequality?
Can you mention all solutions of that inequality?
The solutions can be described on the following number
line.
1
02
-1-3
-4
-2
-5
03
x = 3 on the line is not dotted because 3 is not a solution.
The graph of solution of t
3 is
/ Students Book Linear Equations and Inequalities with One Variable 150
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1
02
-1-3
-4
-2
-5
3
x = 3 in the graph is dotted, because 3 is also a
solution.
Problem 5 Sketch the graph solution of the following inequality
on a number line.
a. y -1 b. m < 5 c. n 0.
Working out an Inequality by Addition or
Division
Look at statement -4 < 1. That statement is true. The
number line below shows what happens if 2 is added to
both sides.
If both sides are added by 2, then we obtain a statement -2
< 3. That statement is also true.
In the example above, adding 2 to both sides does not
change the truth value of the statement.
Now, look at statement -3 < 1. That statement is true.
The line number below shows what happens if 2 is
subtracted from both sides.
1
02
-1-3
-4
-2
-5 4
3
5
+2 +2
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1
02
-1-3
-4
-2
-5 4
3
5
-2 -2
If 2 is subtracted from both sides, then we obtain a
statement -5 < -1. That statement is still true.
In the above example, subtracting 2 from both of sides does
not change the truth of the statement.
Add or subtract a certain number as you wish from
both sides. Are the statements that you have always
true?
Properties of addition or subtraction in an
inequality
If a certain number is added to or subtracted from both sides of
an inequality, the symbol of the inequality does not change, andthe solution does not change, either.The new linear inequality that we get if a certain number isadded to or subtracted from both sides is called a linearinequality equivalent to the original one.
Find the solution set of the followinginequalities:
Example 1
a. y + 2 > 6b. x 3 2, x is an integer between 3 and 8.
Solution :a. y + 2 > 6
y + 2 2 > 6 2 ( 2 is subtracted fromboth sides)
y > 4The graph :
/ Students Book Linear Equations and Inequalities with One Variable 152
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40
5
b. x 3 2
x 3 + 3 2 + 3 ( 3 is subtracted from
both sides)
x 5
Another way:
Because the solutions are not so many, we can check
them one by one.x = -2 (-2) 3 2 x = 3 (3) 3 2-5 2 (true) 0 2 (true)
x = -1 (-1) 3 2 x = 4 (4) 3 2-4 2 (true) 1 2 (true)
x = 0 (0) 3 2 x = 5 (5) 3 2-3 2 (true) 2 2 (true)
x = 1 (1) 3 2 x = 6 (6) 3 2-2 2 (true) 3 2 (false)
x = 2 (2) 3 2 x = 7 (7) 3 2-1 2 (true) 4 2 (false)
Thus, the solution is -2, -1, 0, 1, 2, 3, 4, 5
In your opinion, which way is easier and more
efficient?
Comprehension Check
Find the solution set and sketch the graph of the
solution of the following inequalities.
a. w + 2 > -1
b. 8 1 and the statement 8 < 12.
Those two statements are true. Fill in the blanks below.
First fill it with a suitable number, and then fill it with the
sign or = .
4 > 1
12 = 4 . 3 1. 3 = 3 (both sides are
multiplied by 3)
. . . = 4 . 2 1. 2 = . . . (both sides are
multiplied by 2)
. . . = 4 . 1 1. 1 = . . .(both sides are multiplied by 1)
. . . = 4 . 0 1. 0 = . . (both sides are multiplied by 0)
. . . = 4 . -1 1. -1 = . (both sides are multiplied by -1)
-8 = 4 . -2 1. -2 = -2 (both sides are multiplied by
-2)
. . . = 4 . -3 1. -3 = . . (both sides are multiplied by
-3)
8 < 12
. . . = 8 : 4 12 : 4 = . . . (both sides are divided by 4)
4 = 8 : 2 12 : 2 = 6 (both sides are divided by 2)
. . . = 8 :2
1 12 :
2
1= . . (both sides are divided by
2
1)
-8 = 8 : -1 12 : -1 = -12 (both sides are
divided by -1)
. . . = 8 : -2 12 : -2 = . . (both sides are divided by -2)
/ Students Book Linear Equations and Inequalities with One Variable 154
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. . . = 8 : -4 12 : -4 = . . .(both sides are divided by -4)
Compare the sign in the box that you have filled with
the sign of the beginning statement. What happens if
both sides are multiplied by a positive number, by
zero, or by a negative number? And what happens if
both sides are divided by a positive number, or by a
negative number?
Properties of multiplication or division on bothsides of an inequality
1. if both sides are multiplied or divided by a positive number (non
zero), then the sign of the inequality does not change.2. if both sides are multiplied or divided by a negative number (non
zero), then the sign of the inequality changes into the opposite.
On an inequality:
Example 2 Find the solution set of the following inequalities, and
then sketch the graph of the solution on a number line.
a.2
x < -1.
b. -3
2 x 2.
c. 4x 2 < -2x + 10, x is an integer between -1 and 8
Solution :
a.2
x < -1
2.2
x < 2. 1 (both sides are multiplied by 2, the
sign does not change)
x < -2.
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The graph :
-3
-40
-2
-5
b. -3
2 x 2.
3.(-3
2 x) 3.2 (both sides are multiplied by 3,
the sign does not change)
-2x 6
22
x 26 (both sides are divided by 2, the
sign changes into the opposite)
x -3.
The graph :
-3-4
-5
A car can carry loads not more than 2000 kg. The weight of
the driver and his assistant is 150 kg. He will lift some boxes
of goods. The weight of each box is 50 kg.
a) What is the maximum number of boxes that can be carried in
one route?b) If he carries 350 boxes, what is the minimum number of the
route that must be done?
/ Students Book Linear Equations and Inequalities with One Variable 156
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1. Write an inequality that can state the following
cases.
a) The driver must be 17 years old or more.
b) There are more than 20 species of crocodile.
c) Bus passengers cannot exceed 60 people.
2. Which of the following statements is a linear inequality with
one variable? If the statement is not true, give your reason.
a) 3t + 7 t c) 2m m < 0 c) x x2 > 3
b) y . (y +2) > 2y 1 d) y + y
53. Find the solution set for each inequality, and then sketch the
graph of the solution on a number line.
a) x 1 > 10 f)5
47t -
2
7
b) w + 4 9 g) h -2
1 -1
c) 5 > b 1 h) -74
3 + m +2
1 -24
1
d)2
3+ k -45 i) 3.(v 3) 5 3v
e) 2 < s 8 j)3
4r 3 < r+
3
2-3
1r
4. Critical Thinking Find the value of a so that the
inequality ax + 4 -12 has the solution presented in
the graph below.
-3
-4
-2
-5
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/ Students Book Linear Equations and Inequalities with One Variable 158