powerpoint presentation

1. Subtitle

2. Let’s Prepare
1.
1
6
+
2
6
=
1+2
6
=
3
6
=
𝟏
𝟐
2.
3
2
-
5
2
=
−2
2
= -1

3. Rational Expressions with Like
Denominators
To add or subtract fractions with like denominators simply
add or subtract the numerators. Then write the sum or difference
over the common denominator.
Example 1.
3
𝑥+2
+
1
𝑥+2
=
Since the denominators are the same, that is, both x + 2,
simply add the numerators.
=
3
𝑥+2
+
1
𝑥+2
=
3+1
𝑥+2
=
𝟒
𝒙+𝟐

4. Example 2.
7
𝑚+3
+
2
𝑚+3
=
7+2
𝑚+3
=
𝟗
𝒎+𝟑
Example 3.
4𝑥
𝑥2
−𝑦2 +
4𝑦
𝑥2
−𝑦2 =
4𝑥+4𝑦
𝑥2
−𝑦2 =
4(𝑥+𝑦)
(𝑥+𝑦)(𝑥 −𝑦)
=
𝟒
𝒙 −𝒚
Example 4.
2𝑚
𝑚2 -
3𝑚
𝑚2 =
2𝑚 −3𝑚
𝑚2 =
−𝑚
𝑚2 =
−1 . 𝑚
𝑚 .𝑚
=
−1
𝑚
= -
𝟏
𝒎
Example 5.
𝑚+4
5
+
𝑚 −1
5
=
𝑚+4 +(𝑚 −1)
5
=
𝟐𝒎+𝟑
𝟓

5. Adding and subtracting with like denominators for all rational
numbers, we have
𝑎
𝑐
+
𝑏
𝑐
=
𝑎+𝑏
𝑐
and
𝑎
𝑐
-
𝑏
𝑐
=
𝑎−𝑏
𝑐
.

6. Rational Expressions with Unlike
Denominators
There are instances when rational expressions do not have the
same or like denominators. In these cases, we must first find the
least common denominator (LCD) of the rational expressions given.
LCD is not always recognized, especially in algebraic rational
expressions. It must contain all prime factors of each denominator
raised to the highest power. It is the common multiple of the
denominators.
So, in adding or subtracting rational expressions with unlike
denominators, find first the LCD or the common multiple of the
denominators.

7. Example 1.
3
8
-
1
6
Find the LCD, which is 24.
Solution:
3
8
-
1
6
=
9
24
-
4
24
=
9 −4
24
=
𝟓
𝟐𝟒
Example 2.
𝑦
3
+
2𝑦
4
Solution:
𝑦
3
+
2𝑦
4
=
4𝑦
12
+
6𝑦
12
=
4𝑦+6𝑦
12
=
10𝑦
12
=
𝟓𝒚
𝟔

8. Example 3.
4
3𝑦
-
𝑦+5
5𝑦
Find the LCD:
3y = 3 . y
5y = 5 . y
LCD = y . 3 . 5 = 15y
Solution:
4
3𝑦
-
𝑦+5
5𝑦
=
15𝑦(4)
3𝑦
-
15𝑦(𝑦+5)
5𝑦
=
5 4
15𝑦
-
3(𝑦+5)
15𝑦
=
20
15𝑦
-
3𝑦+15
15𝑦
=
20 −(3𝑦+15)
15𝑦
=
−𝟑𝒚+𝟓
𝟏𝟓𝒚

9. Example 4.
2𝑥+1
9𝑥
-
5
3𝑥2
Find the LCD:
9x = 3 . 3 . x
3x2 = 1 . 3 . x . x
LCD = 3x . 3 . 1 . x = 9x2
So the LCD is 9x2
=
9𝑥2
(2𝑥+1)
9𝑥
-
9𝑥2
(5)
3𝑥2 =
𝑥 2𝑥+1
9𝑥2 -
3(5)
9𝑥2 =
2𝑥2
+𝑥
9𝑥2 -
15
9𝑥2
=
2𝑥2
+𝑥 −15
9𝑥2 or
(𝟐𝒙 −𝟓)(𝒙+𝟑)
𝟗𝒙𝟐

10. When adding or subtracting fractions with unlike denominators,
find first the least common denominators (LCD).
To generalize, we have
𝑎
𝑏
+
𝑐
𝑑
=
𝑎𝑑+𝑏𝑐
𝑏𝑑
→ Addition
𝑎
𝑏
-
𝑐
𝑑
=
𝑎𝑑−𝑏𝑐
𝑏𝑑
→ Subtraction