Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Find the least common multiple of each pair of polynomials.

7. 9 ( x + 2 ) ( 2 x − 1 ) 9 open x plus 2 close open 2 x minus 1 close  and 3 ( x + 2 ) 3 open x plus 2 close
8. x 2 − 1 x squared , minus 1  and x 2 + 2 x + 1 x squared , plus 2 x plus 1
9. 5 y 2 − 80 5 y squared , minus 80  and y + 4 y plus 4
10. x 2 − 32 x − 10 x squared , minus 32 x minus 10  and 2 x + 10 2 x plus 10

See Problems 2 and 3.

Simplify each sum or difference. State any restrictions on the variables.

11. 1 2 x + 1 2 x fraction 1 , over 2 x end fraction , plus , fraction 1 , over 2 x end fraction
12. d − 3 2 d + 1 + d − 1 2 d + 1 fraction d minus 3 , over 2 d plus 1 end fraction . plus . fraction d minus 1 , over 2 d plus 1 end fraction
13. − 2 x − 1 x negative 2 over x , minus , 1 over x
14. − 5 y 2 y − 1 − y + 3 2 y − 1 fraction negative 5 y , over 2 y minus 1 end fraction . minus . fraction y plus 3 , over 2 y minus 1 end fraction
15. 5 y + 2 x y 2 + 2 x − 4 4 x y fraction 5 y plus 2 , over x , y squared end fraction . plus . fraction 2 x minus 4 , over 4 x y end fraction
16. 5 x x 2 − 9 + 2 x + 4 fraction 5 x , over x squared , minus 9 end fraction . plus . fraction 2 , over x plus 4 end fraction
17. y 2 y + 4 − 3 y + 2 fraction y , over 2 y plus 4 end fraction . minus . fraction 3 , over y plus 2 end fraction
18. x 3 x + 9 − 8 x 2 + 3 x fraction x , over 3 x plus 9 end fraction . minus . fraction 8 , over x squared , plus 3 x end fraction
19. − 3 x x 2 − 9 + 4 2 x − 6 fraction negative 3 x , over x squared , minus 9 end fraction . plus . fraction 4 , over 2 x minus 6 end fraction
20. 5 x x 2 − x − 6 + 4 x 2 + 4 x + 4 fraction 5 x , over x squared , minus x minus 6 end fraction . plus . fraction 4 , over x squared , plus 4 x plus 4 end fraction
21. 2 x x 2 − x − 2 − 4 x x 2 + 3 x + 2 fraction 2 x , over x squared , minus x minus 2 end fraction . minus . fraction 4 x , over x squared , plus 3 x plus 2 end fraction

See Problem 4.

Simplify each complex fraction.

22. 1 x 2 y fraction fraction 1 , over x end fraction , over fraction 2 , over y end fraction end fraction
23. 1 − 1 4 2 − 3 5 fraction 1 minus , 1 fourth , over 2 minus , 3 fifths end fraction
24. 2 x + y 3 fraction fraction 2 , over x plus y end fraction , over 3 end fraction
25. 1 3 3 b fraction 1 third , over fraction 3 , over b end fraction end fraction
26. 1 1 + x y fraction 1 , over 1 plus , x over y end fraction
27. 3 2 x + y fraction 3 , over 2 over x , plus y end fraction
28. 2 x + y 5 x + y fraction fraction 2 , over x plus y end fraction , over fraction 5 , over x plus y end fraction end fraction
29. 3 x − 4 1 − 2 x − 4 fraction fraction 3 , over x minus 4 end fraction , over 1 minus . fraction 2 , over x minus 4 end fraction end fraction

    See Problem 5.

30. Your car gets 25 mi/gal around town and 30 mi/gal on the highway.

    1. If 50% of the miles you drive are on the highway and 50% are around town, what is your overall average miles per gallon?
    2. If 60% of the miles you drive are on the highway and 40% are around town, what is your overall average miles per gallon?

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