By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Inequalities By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: August 25, 2007

Solve for x 3x + 5 < 17 - 5 - 5 3x < 12 3 3 x < 4 4 Interval Notation - 5 - 5 3x < 12 3 3 x < 4 4 Jeff Bivin -- LZHS

Solve for x 3x + 5 < 17 - 5 - 5 3x < 12 3 3 x < 4 4 Interval Notation - 5 - 5 3x < 12 3 3 x < 4 4 Jeff Bivin -- LZHS

Solve for x 11 < 3x + 5 < 26 - 5 - 5 - 5 6 < 3x < 21 3 3 3 Interval Notation - 5 - 5 - 5 6 < 3x < 21 3 3 3 2 < x < 7 2 7 Jeff Bivin -- LZHS

Solve for x > > Don't Forget -17 < 5 - 2x < 29 - 5 - 5 - 5 Interval Notation - 5 - 5 - 5 -22 < -2x < 24 > > -2 -2 -2 Don't Forget 11 > x > -12 -12 11 Jeff Bivin -- LZHS

Solve for x Alternative NO YES NO -17 < 5 - 2x < 29 -17 = 5 - 2x Interval Notation -17 = 5 - 2x 5 -2 x = 29 -22 = -2x -2x = 24 11 = x x = -12 -12 11 NO Try -100. Does it work? YES NO Try 100. Does it work? Try 0. Does it work? Jeff Bivin -- LZHS

Solve for x Alternative NO YES NO 11 < 5 + 2x < 19 11 = 5 + 2x Interval Notation 11 = 5 + 2x 5 + 2 x = 19 6 = 2x 2x = 14 3 = x x = 7 11 < 5 + 2(0) < 19 3 7 NO 11 < 5 + 2(5) < 19 Try 0. Does it work? YES NO Try 10. Does it work? Try 5. Does it work? 11 < 5 + 2(10) < 19 Jeff Bivin -- LZHS

| 2x + 3 | < 9 NO YES NO 2x + 3 = 9 2x + 3 = -9 2x = 6 2x = -12 Interval Notation 2x + 3 = 9 2x + 3 = -9 2x = 6 2x = -12 x = 3 x = -6 | 2(-10) + 3 | < 9 -6 3 NO Try -10. Does it work? YES NO | 2(0) + 3 | < 9 Try 100. Does it work? Try 0. Does it work? | 2(100) + 3 | < 9 Jeff Bivin -- LZHS

| 2x - 7 | > 11 YES NO YES 2x - 7 = 11 2x - 7 = -11 2x = 18 2x = -4 Interval Notation 2x - 7 = 11 2x - 7 = -11 2x = 18 2x = -4 x = 9 x = -2 | 2(-10) - 7 | > 11 -2 9 YES Try -10. Does it work? NO YES | 2(0) - 7 | > 11 Try 10. Does it work? Try 0. Does it work? | 2(10) - 7 | > 11 Jeff Bivin -- LZHS

Absolute value is always positive | 5x + 2 | > -4 Absolute value is always positive Isn’t a positive number always bigger than a negative number? YES! Therefore, any number will work. Interval Notation Jeff Bivin -- LZHS

1 < | x - 2 | < 5 -3 1 3 7 Interval Notation: | x - 2 | = 1 Try -10 Try Try 2 Try 5 Try 10 1 < | -10 - 2 | < 5 1 < | 0 - 2 | < 5 1 < | 2 - 2 | < 5 1 < | 5 - 2 | < 5 1 < | 10 - 2 | < 5 Interval Notation: 1 < | -12 | < 5 1 < | -2 | < 5 1 < | 0 | < 5 1 < | 3 | < 5 1 < | 8 | < 5 1 < 12 < 5 1 < 2 < 5 1 < 0 < 5 1 < 3 < 5 1 < 8 < 5 Jeff Bivin -- LZHS

5 < | x + 9 | < 11 2 -20 -14 -4 Interval Notation: | x + 9 | = 5 Try -30 Try -15 Try -10 Try Try 10 5 < | -30 + 9 | < 11 5 < | -15 + 9 | < 11 5 < | -10 + 9 | < 11 5 < | 0 + 9 | < 11 5 < | 10 + 9 | < 11 Interval Notation: 5 < | -21 | < 11 5 < | -6 | < 11 5 < | -1 | < 11 5 < | 9 | < 11 5 < |19 | < 11 5 < 21 < 11 5 < 6 < 11 5 < 1 < 11 5 < 9 < 11 1 < 19 < 5 Jeff Bivin -- LZHS

-2 < | x + 1 | < 5 -6 4 Interval Notation: | x + 1 | = 5 A negative is always less than an absolute value! x + 1 = 5 x + 1 = -5 Therefore, we have NO partitions here. x = 4 x = -6 -6 4 Try -10 Try Try 10 -2 < | -10 + 1 | < 5 -2 < | 0 + 1 | < 5 -2 < | 10 + 1 | < 5 Interval Notation: -2 < | -9 | < 5 -2 < | 1 | < 5 -2 < | 11 | < 5 -2 < 9 < 5 -2 < 1 < 5 -2 < 11 < 5 Jeff Bivin -- LZHS

7 < | 2x + 11 | < 3 ? What number is bigger than 7 and smaller than 3 at the same time? There isn’t any! Ø Jeff Bivin -- LZHS

(x + 3)(x – 4)(x + 1)(x – 2) > 0 Try -10 -3 Try -2 -1 Try 2 Try 3 4 Try 10 (x + 3) (x – 4) (x + 1) (x – 2) > 0 x = -10: ( – ) ( – ) ( – ) ( – ) = + > 0 ? YES x = -2: ( + ) ( – ) ( – ) ( – ) = – > 0 ? NO Interval Notation: x = 0: ( + ) ( – ) ( + ) ( – ) = + > 0 ? YES x = 3: ( + ) ( – ) ( + ) ( + ) = – > 0 ? NO x = 10: ( + ) ( + ) ( + ) ( + ) = + > 0 ? YES Jeff Bivin -- LZHS