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Solving Rational Equations and Inequalities
Objective: Use algebra to solve rational inequalities.
43c 2
15r
1,
)1)(43(043
34
2
x
xxxx4,9);4)(9( xxx
43c 2
15r
4,9);4)(9( xxx
6.9;485 mm1,
)1)(43(043
34
2
x
xxxx
24;605.2 gg
2.1;1210 aa
29
31
211
;29
236
666
xx
x
xxx xx
Example 1
Example 1
Example 1
Example 1
Example 2
Example 2
Example 2
Try This
• Solve2
1
3 x
x
Try This
• Solve2
1
3 x
x
1,3
0)1)(3(
032
3)2(2
x
xx
xx
xx
true;1123
1
3
3
true;3
1
3
121
1
3
1
Try This
• Solve2
1
3 x
x
1,3
0)1)(3(
032
3)2(2
x
xx
xx
xx
true;1123
1
3
3
true;3
1
3
121
1
3
1
Extraneous Solutions
• Sometimes solving a rational equation introduces extraneous solutions. An extraneous solution is a solution to a resulting equation that is not a solution to the original equation. Therefore, it is important to check your answers.
Example 3
Example 3
Example 3
Example 3
Try This
• Solve65
3
32 2
xxx
x
x
x
Try This
• Solve65
3
32 2
xxx
x
x
x
65
3)3)(2(
3)3)(2(
2)3)(2(
2
xxxx
x
xxx
x
xxx
Try This
• Solve65
3
32 2
xxx
x
x
x
65
3)3)(2(
3)3)(2(
2)3)(2(
2
xxxx
x
xxx
x
xxx
3,
0)3)(12(
0352
0323
3)2()3(
21
2
22
xx
xx
xx
xxxx
xxxx
Try This
• Solve65
3
32 2
xxx
x
x
x
65
3)3)(2(
3)3)(2(
2)3)(2(
2
xxxx
x
xxx
x
xxx
3,
0)3)(12(
0352
0323
3)2()3(
21
2
22
xx
xx
xx
xxxx
xxxx
solution a is
solution anot is 3
21
x
x
Try This
• Solve65
3
32 2
xxx
x
x
x
21
2;0
65
3
32
x
xxx
x
x
x
Example 4
Example 4
Example 4
Try This
• Solve 32
1
x
x
Try This
• Solve
• If x+2 > 0, then x > -2
32
1
x
x
2
72
631
)2(31
27
x
x
x
xx
xx
Try This
• Solve
• If x+2 > 0, then If x + 2 < 0, then x > -2 x < -2
32
1
x
x
2
72
631
)2(31
27
x
x
x
xx
xx
27
72
631
)2(31
x
x
xx
xx
Try This
• Solve 32
1
x
x
2,27 xx
Example 5
Example 5
Example 5
Example 5
Homework
• Pages 517-518• 9-27 odd
