Pre-CalculusLesson 7: Solving InequalitiesLinear inequalities, compound inequalities, absolute value inequalities, interval notation

Number Line

< and > are shown with open circles

x<2 x>4

Number Line

< and > are shown with open circles

x<2 x>4

Number Line

< and > are shown with open circles

x<2 x>4

Number Line

and are shown with closed circles

x 2 x 4

Number Line

and are shown with closed circles

x 2 x 4

Number Line

and are shown with closed circles

x 2 x 4

Multiplication Property of Inequality

When multiplying or dividing by a negative number, FLIP the INEQUALITY SIGN!

Example:

756 y

Example:

756 y66

Example:

756 y66 15 y

Example:

756 y66 15 y

55

Example:

756 y66 15 y

55

5

1y

Compound Inequalities

Conjunction Example #17432 x

-3 -2 -1 0 1 2

Conjunction Example #17432 x

444

-3 -2 -1 0 1 2

Conjunction Example #17432 x

444 336 x

-3 -2 -1 0 1 2

Conjunction Example #17432 x

444 336 x333

12 x

-3 -2 -1 0 1 2

Conjunction Example#2634 x

6 7 8 9 10 11

Conjunction Example#2634 x333

97 x

6 7 8 9 10 11

Conjunction Example#2634 x333

97 x111

97 x

6 7 8 9 10 11

Disjunction Example#13434 xorx

0 1 2 3 4 5 6 7 8 9 10

Disjunction Example#13434 xorx4444

71 xorx

0 1 2 3 4 5 6 7 8 9 10

Disjunction Example#235342 xorx

-5 -4 -3 -2 -1 0 1 2 3 4 5

Disjunction Example#235342 xorx

5544 272 xorx

-5 -4 -3 -2 -1 0 1 2 3 4 5

Disjunction Example#235342 xorx

5544 272 xorx

-5 -4 -3 -2 -1 0 1 2 3 4 5

22

22

7 xorx

Absolute Value Inequalities

“Less Than”

Rewrite the inequality as a conjunction.

-a < x < a

Solve.

-4 -3 -2 -1 0 1 2

Example 743 x

-4 -3 -2 -1 0 1 2

Example

7437 x

743 x

-4 -3 -2 -1 0 1 2

Example

7437 x444

3311 x

743 x

-4 -3 -2 -1 0 1 2

Example

7437 x444

3311 x333

13

11 x

743 x

“Greater Than”

Rewrite the inequality as a disjunction.

x < -a or x > a

Solve.

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

342 x

Example342342 xorx

-5 -4 -3 -2 -1 0 1 2 3 4 5

342 x

Example342342 xorx

4444 1272 xorx

-5 -4 -3 -2 -1 0 1 2 3 4 5

342 x

Example342342 xorx

4444 1272 xorx

-5 -4 -3 -2 -1 0 1 2 3 4 5

2222

2

1

2

7 xorx

342 x

Interval Notation When using interval notation:

( means "not included" or "open". [ means "included" or "closed".

The inequality would be written as the interval The inequality would be written as the interval

62 x 6,232 xorx

,32,

Which statement below is the correct interval notation for the situation depicted in this number line graph?

http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm

Which statement below is the correct interval notation for the situation depicted in this number line graph?

http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm

Write the following statement as an inequality:

x < -3   or   0 < x < 2    or    x > 4 x < -3   or   0 < x < 2    or    x > 4 x < -3   or   0 < x < 2    or    x > 4

http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm

Write the following statement as an inequality:

x < -3   or   0 < x < 2    or    x > 4 x < -3   or   0 < x < 2    or    x > 4 x < -3   or   0 < x < 2    or    x > 4

http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm

Write the following inequality as interval notation: -2 < x < 1  or  x > 1

http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm

Write the following inequality as interval notation: -2 < x < 1  or  x > 1

http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm

Practice QuestionsSolve each inequality, express the answer in interval notation, and graph the solution on the number line.1.

2.

3.

4.

5.