Lecture 8 - Equations involving Rational Expressions...

Equations involving Rational Expressions, Radicals,
Absolute Values, and those in Quadratic Form
Mathematics 17
Institute of Mathematics, University of the Philippines-Diliman
Lecture 8
Math 17 (UP-IMath) Equations Lec 8 1 / 19

Outline
1
Rational Equations in x
2
Equations involving Radicals
3
Equations in Quadratic Form
4
Equations Involving Absolute Values
5
Solving Equations via Factoring
Math 17 (UP-IMath) Equations Lec 8 2 / 19

Equations involving Rational Expressions
To solve:
Math 17 (UP-IMath) Equations Lec 8 3 / 19

Equations involving Rational Expressions
To solve:
1 Multiply the LCD to both sides of the equation
2 Solve the equation that arises
3 Drop any of the obtained solutions that make the rational expressions
undefined (extraneous solutions).
Math 17 (UP-IMath) Equations Lec 8 3 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
( x - 5)( x + 2)
h
x +4
x +2
-
2 x - 3
x - 5
i
=
3 x - 8
x 2 - 3 x - 10
· ( x - 5)( x + 2)
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
( x - 5)( x + 2)
h
x +4
x +2
-
2 x - 3
x - 5
i
=
3 x - 8
x 2 - 3 x - 10
· ( x - 5)( x + 2)
= 3 x - 8
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
( x - 5)( x + 2)
h
x +4
x +2
-
2 x - 3
x - 5
i
=
3 x - 8
x 2 - 3 x - 10
· ( x - 5)( x + 2)
( x - 5)( x + 4) = 3 x - 8
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
( x - 5)( x + 2)
h
x +4
x +2
-
2 x - 3
x - 5
i
=
3 x - 8
x 2 - 3 x - 10
· ( x - 5)( x + 2)
( x - 5)( x + 4) - (2 x - 3)( x + 2) = 3 x - 8
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
( x - 5)( x + 2)
h
x +4
x +2
-
2 x - 3
x - 5
i
=
3 x - 8
x 2 - 3 x - 10
· ( x - 5)( x + 2)
( x - 5)( x + 4) - (2 x - 3)( x + 2) = 3 x - 8
= 3 x - 8
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
( x - 5)( x + 2)
h
x +4
x +2
-
2 x - 3
x - 5
i
=
3 x - 8
x 2 - 3 x - 10
· ( x - 5)( x + 2)
( x - 5)( x + 4) - (2 x - 3)( x + 2) = 3 x - 8
x
2
- x - 20 = 3 x - 8
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
( x - 5)( x + 2)
h
x +4
x +2
-
2 x - 3
x - 5
i
=
3 x - 8
x 2 - 3 x - 10
· ( x - 5)( x + 2)
( x - 5)( x + 4) - (2 x - 3)( x + 2) = 3 x - 8
x
2
- x - 20 - (2 x
2
+ x - 6) = 3 x - 8
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
( x - 5)( x + 2)
h
x +4
x +2
-
2 x - 3
x - 5
i
=
3 x - 8
x 2 - 3 x - 10
· ( x - 5)( x + 2)
( x - 5)( x + 4) - (2 x - 3)( x + 2) = 3 x - 8
x
2
- x - 20 - (2 x
2
+ x - 6) = 3 x - 8
x
2
- x - 20 - 2 x
2
- x + 6 = 3 x - 8
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.
Solution:
( x - 5)( x + 2)
h
x +4
x +2
-
2 x - 3
x - 5
i
=
3 x - 8
x 2 - 3 x - 10
· ( x - 5)( x + 2)
( x - 5)( x + 4) - (2 x - 3)( x + 2) = 3 x - 8
x
2
- x - 20 - (2 x
2
+ x - 6) = 3 x - 8
x
2
- x - 20 - 2 x
2
- x + 6 = 3 x - 8
- x
2
- 5 x - 6 = 0
Math 17 (UP-IMath) Equations Lec 8 4 / 19

Example: Solve
x +4
x +2
-
2 x - 3
x - 5
=
3 x - 8
x 2 - 3 x - 10
.