2.6 Other Types of Equations

* Solve equations involving rational exponents
* Solve equations using factoring
* Solve equations with radicals and check the solutions
* Solve absolute value equations
* Solve other types of equations

1. 2.6 Other Types of Equations
Chapter 2 Equations and Inequalities

2. Concepts and Objectives
⚫Objectives for this section are:
⚫Solve equations involving rational exponents
⚫Solveequationsusingfactoring
⚫Solve equations with radicals and check the solutions
⚫Solve absolute value equations
⚫Solveothertypesofequations

3. RationalExponentEquations
⚫Recall that a rational exponent indicates a power in the
numerator and a root in the denominator.
⚫Therearemultiplewaysofwritinganexpressionwitha
rationalexponent:
⚫If we are given an equation in which a variable is raised
to a rational exponent, the simplest way to remove the
exponent on xis by raising both sides of the equation to
a power that is the reciprocal of the exponent.() ()
/1/
mm
mnn m n n
aaaa===

4. Rational Exponent Equations
⚫Example: Solve
The reciprocalofis , so 5/4
32x= 5
4 4
5 ()()
4/5 4/5
5/4
32
16
x
x
=
=

5. Rational Exponent Equations
⚫Example: Solve
Wecannowusethezero-factorproperty:3/41/2
3xx= ( )
3/41/2
3/42/4
2/41/4
30
30
310
xx
xx
xx
−=
−=
−=

6. Rational Exponent Equations
⚫Example: Solve
Solutionsare0and3/41/2
3xx= 2/4
0
0
x
x
=
= 1/4
1/4
1/4
4
310
31
1
3
11
381
x
x
x
x
−=
=
=

==

 1
81

7. Solving by Grouping
⚫If a polynomial consists of 4 terms, we can solve it by
grouping.
⚫Groupingproceduresrequirefactoringthefirsttwo
terms,andthenfactoringthelasttwoterms. If the
factors in the parentheses are identical, then the
expression can be factored by grouping.

8. Solving by Grouping
⚫Example: Solve
Solutions are ‒1, 3, ‒332
990xxx+−−= ()()
()( )
()()()
32
2
2
990
1910
190
1330
xxx
xxx
xx
xxx
+−−=
+−+=
+−=
+−+=
difference of squares!

9. Power Property
⚫Note:This does notmean that every solution of P
n
= Q
n
is a solution of P= Q.
⚫We use the power property to transform an equation
that is difficult to solve into one that can be solved more
easily. Whenever we change an equation, however, it is
essentialto check all possible solutions in the original
equation.
If Pand Qare algebraic expressions, then every
solution of the equation P= Qis also a solution of
the equation P
n
= Q
n
, for any positive integer n.

10. Solving Radical Equations
⚫Step 1Isolate the radical on one side of the equation.
⚫Step 2Raise each side of the equation to a power that is
the same as the index of the radical to eliminate the
radical.
⚫If the equation still contains a radical, repeat steps 1
and 2.
⚫Step 3Solve the resulting equation.
⚫Step 4Check eachproposed solution in the original
equation.

11. Solving Radical Equations (cont.)
⚫Example: Solve − + =4 12 0xx

12. Solving Radical Equations (cont.)
⚫Example: Solve − + =4 12 0xx =+4 12xx =+
2
4 12xx − − =
2
4 12 0xx ()()− + =6 2 0xx =−6, 2x

13. Solving Radical Equations (cont.)
⚫Example: Solve
Check:
Solution: {6}− + =4 12 0xx 412xx=+ 2
412xx=+ 2
4120xx−−= ()()620xx−+= 6,2x=− ()− + =6 4 6 12 0 −=6 36 0 −=6 6 0 =00 ()− − − + =2 4 2 12 0 − − =2 4 0 − − =2 2 0 −40

14. Solving Radical Equations (cont.)
⚫Example: Solve + − + =3 1 4 1xx

15. Solving Radical Equations (cont.)
⚫Example: Solve + − + =3 1 4 1xx ( )
2
41x++ ()
2
2
3141
314241
2424
24
444
50
50
0, 5
xx
xxx
xx
xx
xxx
xx
xx
x
+=++
+=++++
−=+
−=+
−+=+
−=
−=
=

16. Solving Radical Equations (cont.)
⚫Example: Solve
Check:
Solution: {5}+ − + =3 1 4 1xx ()
2
2
3141
314241
2424
24
444
50
50
0, 5
xx
xxx
xx
xx
xxx
xx
xx
x
+=++
+=++++
−=+
−=+
−+=+
−=
−=
= ()+ − + =3 0 1 0 4 1 −=1 4 1 −=1 2 1 −11 ()+ − + =3 5 1 5 4 1 −=16 9 1 −=4 3 1 =11

17. Absolute Value Equations
⚫You should recall that the absolute valueof a number a,
written |a|, gives the distance from ato 0 on a number
line.
⚫By this definition, the equation |x| = 3 can be solved by
finding all real numbers at a distance of 3 units from 0.
Both of the numbers 3 and ‒3 satisfy this equation, so
the solution set is {‒3, 3}.

18. Absolute Value Equations (cont.)
⚫The solution set for the equation must include
both aand –a.
⚫Example: Solve =xa −=9 4 7x

19. Absolute Value Equations (cont.)
⚫The solution set for the equation must include
both aand –a.
⚫Example: Solve
The solution set is =xa −=9 4 7x −=9 4 7x − =−9 4 7x − =−42x − =−4 16x =
1
2
x =4x
or


1
,4
2

20. Quadratic in Form
⚫An equation is said to be quadratic in formif it can be
written as
where a0 and uis some algebraic expression.
⚫To solve this type of equation, substitute ufor the
algebraic expression, solve the quadratic expression for
u, and then set it equal to the algebraic expression and
solve for x. Because we are transforming the equation,
you will still need to check any proposed solutions against
the original equation.+ + =
2
0au bu c

21. Quadratic in Form (cont.)
⚫Example: Solve ()()− + − − =
2 3 1 3
1 1 12 0xx

22. Quadratic in Form (cont.)
⚫Example: Solve
Let This makes our equation:()()− + − − =
2 3 1 3
1 1 12 0xx ()=−
13
1ux + − =
2
12 0uu ()()+ − =4 3 0uu =−4, 3u

23. Quadratic in Form (cont.)
⚫Example: Solve
Let . This makes our equation:
So, and ()()− + − − =
2 3 1 3
1 1 12 0xx ()=−
13
1ux + − =
2
12 0uu ()()+ − =4 3 0uu =−4, 3u ()
() ()
13
3
1/3 3
14
14
164
63
x
x
x
x
−=−
−=−

−=−
=− ()
() ()
13
3
1/3 3
13
13
127
28
x
x
x
x
−=
−=

−=
=

24. Quadratic in Form (cont.)
⚫Example: Solve (cont.)
Now, we have to check our proposed solutions:()()− + − − =
2 3 1 3
1 1 12 0xx ( )( )− − + − − − =
2 3 1 3
63 1 63 1 12 0 ()()− + − − =
2 3 1 3
64 64 12 0 ()()− + − − =
21
4 4 12 0 − − =16 4 12 0 =00

25. Quadratic in Form (cont.)
⚫Example: Solve (cont.)
Solution: {–63, 28}()()− + − − =
2 3 1 3
1 1 12 0xx ( )( )− + − − =
2 3 1 3
28 1 28 1 12 0 ()()+ − =
2 3 1 3
27 27 12 0 ()()+ − =
21
3 3 12 0 + − =9 3 12 0 =00

26. Rational Equations (Again)
⚫A rational equationis an equation that has a rational
expression for one or more terms.
⚫To solve a rational equation, multiply both sides by the
lowest common denominator of the terms of the
equation. Be sure to check your solution against the
undefined values!
Because a rational expression is not defined when its
denominator is 0, any value of the variable which makes
the denominator’s value 0 cannot be a solution.

27. Rational Equations (cont.)
⚫Example: Solve 2
3212
22
x
xxxx
+−
+=
−−

28. Rational Equations (cont.)
⚫Example: Solve
The LCD is xx‒ 2, which means x0, 2.2
3212
22
x
xxxx
+−
+=
−− () () ()
()2
22
3212
2
2
x
xxxx
xxxxxx
+−
+=  
−
−−−
− 

29. Rational Equations (cont.)
⚫Example: Solve
The LCD is xx‒ 2, which means x0, 2.2
3212
22
x
xxxx
+−
+=
−− () () ()
()
( )()
()
2
2
3212
22
3222
3222
2
330
310
22
0,1
x
x
xxxx
xxx
x
xxxxx
xx
xx
xx
x
+−
+=  
−− 
++−=−
++−=−
+=
+=
=
−−
−
− 1−

30. Classwork
⚫College Algebra2e
⚫2.6: 6-18 (even); 2.5: 20-30 (even); 2.4: 28-40 (even)
⚫2.6 Classwork Check
⚫Quiz 2.5