|
|
|
|
|
|
Linear
equations |
Linear
equation |
|
Solving linear equations |
|
Applications
of linear equations, word problems |
|
Numbers word problem |
|
Geometry and physics word problems
|
|
|
Linear inequalities |
|
Inequality signs
|
|
Properties
of inequalities
|
|
Solving
linear inequalities |
|
|
|
|
|
|
| Linear
equation |
| An equation is linear if it can be written in the form |
| ax =
b,
where
x
is the variable and a and
b are constants. |
| A linear, means the
variable x appears only to the first power. |
| The solution of an
equation is the value of the variable x that makes the equation a
true statement. |
|
| Solving linear equations |
| We solve an equation by transforming it through a
sequence of equivalent equations, statement by statement, until x
finally is isolated on the left. |
| Two equations that have the same
solutions are called equivalent. |
| Shift a number from one side of an
equation to the other, by writing it on the other side with the inverse
operation. |
| When the operations are addition or subtraction a term
may be shifted to the other side of an equation by changing its sign. |
| After solving an equation check the solution by substituting them in
the original equation. |
|
| Examples:
a)
-1
-
(2
-
x)
= 2
-
(3x
+
1) |
|
-1
-
2
+
x
= 2
-
3x
-
1 |
|
x
+
3x
= 2 +
2 |
|
4x
= 4
|
¸
4 |
|
x
= 1 |
|
|
|
|
| Applications of the linear
equations,
word
problems
|
| Word problems - translate the words into mathematics |
| Numbers word
problem |
| Example:
The sum of two digits of a two-digit number is 10. If the digits are
reversed, the new number is |
|
four less then five times the original number. Find two-digit number. |
| Solution:
Let
x be ten's digit then
10
-
x is units digit, thus given number is
10x
+ (10
-
x). |
| By reversing digits we get
10
· (10
-
x)
+ x. |
| Hence
given problem yields: 10
· (10
-
x)
+ x
= 5
· [10x
+ (10
-
x)]
-
4 |
|
100
-
10x
+ x
= 5
· [9x
+ 10]
-
4 |
|
100
-
9x
= 45x
+ 50
-
4 |
|
54x
= 54
| ¸
54 |
|
x
= 1
thus,
the two-digit number is:
10x
+ (10
-
x)
=19. |
|
| Geometry and physics word problems
|
| Example:
A rectangle has the perimeter
90 cm and the width 13 cm less then the length. |
|
What is the area of the rectangle? |
|
|
 |
|
a
= b
+ 13
= 16
+
13
= 29
cm |
| A
= a
· b
= 29
·
16
= 464
cm2 |
|
| the perimeter
P
of a rectangle: |
| P
= 2
·
(a
+ b)
= 90 |
|
2
·
(b + 13
+ b)
= 90
| ¸
2 |
|
2b
+ 13
= 45 |
|
2b
= 32 |
|
b
= 16
cm |
|
|
|
| Example:
After a car A started traveling at a speed of 90 km/h, half-hour later
from the same place in the |
|
same direction starts car B traveling at 120 km/h. When will car B reach the car A? |
|
|
Solution:
The cars are traveling the same distance
d
= r
· t .
distance = rate x time |
|
The car A takes x hours, and the car B,
x
- 1/2
hours. |
|
Thus, 90
· x =
120
· (x
- 1/2)
| ¸
30 |
|
3x
=
4x
- 2 |
|
x
=
2
hours, the car B will reach the car A after
x
- 1/2 =
2
- 1/2 =
3/2 hours. |
|
|
|
| Linear
inequalities |
|
Inequality signs
|
|
Properties
of inequalities
|
|
Solving
linear inequalities |
|
| Inequality signs
|
| Inequality
is a statement about the relationship between two numbers or
quantities that holds when they are |
| comparable and related by
ordering but are not equal. |
| These
relationships are written as: |
|
a
< b
meaning
a
is
less than b, |
|
a ≤
b
meaning
a
is
less than or equal to b, |
|
a
> b
meaning
a
is
greater than b, |
|
a ≥
b
meaning
a
is
greater than or equal to b. |
|
| Properties
of inequalities
|
| The
trichotomy property of the real
line states that for any real numbers, a
and
b,
exactly one of |
|
a
< b,
a = b,
a
> b,
is true. |
|
| The
transitive property of inequalities
states that for any real numbers, a,
b and
c, |
|
if a
< b
and b
< c,
then
a
< c, |
|
if a
> b and
b
> c,
then
a
> c. |
|
| The
addition and subtraction property
of inequalities states that for any real numbers, a,
b and
c, |
|
if a
< b,
then
a
+ c
< b
+ c,
and a
-
c
< b
-
c, |
|
if a
> b,
then
a
+ c
> b
+ c, and
a
-
c
> b
-
c. |
|
| The
multiplication and division
property of inequalities states that for any real numbers, a,
b and
c, |
|
if c
is
positive and
a
< b,
then
a
· c
< b
· c,
and a/c
< b/c, |
|
if c
is
negative and
a
< b,
then
a
· c
> b
· c,
and a/c
> b/c. |
|
| The
sense of an inequality is not changed if both sides are
increased or decreased by the same number, or if |
| both sides are
multiplied or divided by a positive number. |
| The sense of an
inequality is reversed if both sides are multiplied or divided
by a negative number. |
|
| The
additive inverse property of inequalities states that for any real
numbers a
and
b, |
|
if
a
< b
then
-a
>
-b, |
|
if
a > b
then
-a
< -b. |
|
| The
multiplicative inverse
property of inequalities states that for any real numbers a
and
b
that are both |
|
positive or both negative, |
|
if
a
< b
then
1/a
>
1/b, |
|
if
a > b
then
1/a
<
1/b. |
|
| Note that, the above properties also hold if strict inequality signs, < and >, are replaced with their |
| corresponding non strict
inequality signs, ≤
and ≥. |
|
| Solving
linear inequalities
|
| A linear inequality has the standard form
ax +
b
> 0, where a,
b
Î
R |
| (and where other inequality
signs like <, > and < can appear). |
| As with equations, the inequality is solved when positive x is isolated on the left. |
| The solutions to an inequality are all values of
x
that make the inequality true. |
| We may add the same number to both sides of an inequality. |
| By multiplying both sides of an inequality
by the same negative number, the sense of the inequality changes, |
|
i.e., it reverses the direction of the inequality sign. |
|
| Example: |
 |
|
x -
8
< 2x
-
4 |
|
- x
<
4
|
·
(-1) |
|
x
>
-4 |
|
 |
| The
solution of the inequality is every real number greater
than -4. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Beginning
Algebra Contents |
|
|
 |
|
| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
|
|
