Interval Notation / Absolute Value Worksheet

Interval Notation / Absolute Value Worksheet Answers

Graph the given intervals and find their intersection.

1. ( -3, 2) and [ 1 , 4 ) intersection = [ 1 , 2 )

-3 1 2 4

2. ( - ∞, 3 ) and ( - 2 , + ∞ ) intersection = ( - 2 , 3 )

Solve and graph each inequality.

3.  6x  2  5 7  6x  2  5  6  2  2  6x  7 Divided by a negative, inequality  6x 7  switches direction  6  6 7 x   6

7 4. 1  2x  3  4 x  1 , x  2 1  2x  3  4  3   3   3 2  2x  7 2 2x 7 1 7   2 2 2 2 7 1  x  2  1  5. x 1 x    0  2  TTTT FFFFFFFFF TTTT Critical points are … 1 1 1  x 1  0 and x   0 2 2 1 1 1  1    2 2 1 x  1 and x   2

TEST x = 0

 1  0 10    0  2   1  1   0  2  1   0 2

 2  1  6. x x   x    0  3  3  Critical points are … 2 1 x  0 and x   0 and x   0 3 3 2 1 x  0, x  , x   3 3

TEST x = 1 TTTT FF TTT FFFF  2  1  11 1   0 1 2 3 3  0    3 3  1  4  1    0  3  3  4  0 FALSE 9 7. 4x3  6x 2  0 Critical points are … FFFF TTTT FFFF 2x 2 2x  3  0 0 3

2 2 2x  0 2x  3  0 3 x  0 x  2

TEST x = 1

413  612  0 4  6  0 TRUE  2  0

Solve the absolute value equations.

8. 6x  5  0

6x  5  0  5  5 6x  5 5 x   6

  5  6  5  0  6   5  5  0 0  0 0  0 2 9. x 1  3 x 1  4  0 Let u = x 1

u 2  3u  4  0 u  4u 1  0

x 1  4  0 x 1  4

No solution, abs. value ≠ (-)

x 1 1  0 x 1  1

1  x 1  1 1  x 1  1 Solution set x = { 0 , - 2 } 1  1  1 - 2  x  0

0 1 2  30 1  4  0 1 2  31  4  0 1 3  4  0 0  0

 2 1 2  3 2 1  4  0 1 2  31  4  0 1 3  4  0 0  0 Solve and find the solution for each absolute value inequality as an interval.

10. x  2  1

1  x  2  1  2   2  2 1 3 1  x  3 Interval ( 1 , 3 )

11. x  3  3

 3  x  3  3 - 6 0  3   3  3  6  x  0 Interval ( - ∞ , -6 ] U [ 0 , + ∞ )

12. 2x 1  1

1  2x 1  1 - 1 0 1  1  1  2  2x  0 Interval ( - ∞ , -1 ] U [ 0 , + ∞ )  2 2x 0   2 2 2 1  x  0

1 2 13.  2x   3 3 1 1  2 6 2 1 2   2x   3 3 3 1 1 1      3 3 3  1   1  Interval  ,    , 1 2 6   2x  1     3 1 1  x   6 2