7. First, consider the case where 4 − x > 0 , 4 minus x greater than 0 comma  or x < 4 . x less than 4 .  Justify each step.

   Hint: The solution must satisfy both x < 3 x less than 3  and x < 4 . x less than 4 .

   x 4 − x < 3 fraction x , over 4 minus x end fraction . less than 3

   1. ( 4 − x ) x 4 − x < ( 4 − x ) 3 open , 4 minus x , close . fraction x , over 4 minus x end fraction . less than . open , 4 minus x , close . 3
   2. x < 12 − 3 x x less than , 12 minus 3 x
   3. 4 x < 12 4 x less than 12
   4. x < 3 x less than , 3
8. Combine this result with the given condition x < 4 . x less than 4 .  What is the solution for this case?
9. Now consider the case where 4 − x < 0 , 4 minus x less than 0 comma  or x > 4 . x greater than 4 .  Justify each step.

   Hint: The solution must satisfy both x > 3 x greater than 3  and x > 4 . x greater than 4 .

   x 4 − x < 3 fraction x , over 4 minus x end fraction . less than 3

   1. ( 4 − x ) x 4 − x > ( 4 − x ) 3 open , 4 minus x , close . fraction x , over 4 minus x end fraction . greater than . open , 4 minus x , close . 3
   2. x > 12 − 3 x x greater than , 12 minus 3 x
   3. 4 x > 12 4 x greater than 12
   4. x > 3 x greater than , 3
10. Combine this result with the given condition x > 4 . x greater than 4 .  What is the solution for this case?
11. Examine your solutions for Exercises 8 and 10. Now, write the solution of the inequality x 4 − x < 3 . fraction x , over 4 minus x end fraction . less than 3 .

Exercises

For Exercises 12-17, solve each inequality graphically and algebraically.

12. 2 x − 1 < x fraction 2 , over x minus 1 end fraction . less than x
13. x + 1 > x + 5 x + 2 x plus 1 greater than . fraction x plus 5 , over x plus 2 end fraction
14. 2 x ( x − 2 ) ( x + 3 ) < 1 fraction 2 x , over open , x minus 2 , close . open , x plus 3 , close end fraction . less than 1
15. 2 x + 2 x − 1 < x + 1 fraction 2 x plus 2 , over x minus 1 end fraction . less than x plus 1
16. x 2 + 1 x < 2 x fraction x squared , plus 1 , over x end fraction . less than 2 x
17. x − 1 x − 2 < x + 3 x − 1 fraction x minus 1 , over x minus 2 end fraction . less than . fraction x plus 3 , over x minus 1 end fraction

18. For Exercises 12-17, check your work by using a table to solve each inequality.

    The equation d = rt relates the distance d you travel, the time t it takes to travel that distance, and the rate r at which you travel. So the time it takes to travel a distance d at a rate r is t = d r . t equals , d over r , .  If you increase your rate by a to r + a, then it takes less time, t = d r + a . t equals . fraction d , over r plus eh end fraction . .  In fact, the time you save by going at the faster rate is T = d r − d r + a . t equals , d over r , minus . fraction d , over r plus eh end fraction . .

19. 1. You normally take a 500-mi trip, averaging 45 mi/h. You want to increase the rate so that you save at least an hour. Write an inequality that describes the situation.
    2. Solve your inequality from part (a).

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