3-4 Extension Compound Inequalities

Compound Inequalities•Formed by joining two inequalities with the word “and” or “or”. •Examples:

and or

Example 1:•Write as a single inequality and then graph:•x > -8 and x ≤ 4

Example 2:•Graph y ≤ 0 or y ≥ 7

You Try!•Graph each compound inequality:•k > 3 and k < 9

•w < -10 or w ≥ -6

Writing From Graphs•Write a compound inequality for:

Example•Write a compound inequality for:

You Try!•Write an inequality for:

Solving with “and”•Solve like an equation, but do each thing to all 3 parts!•Example: •4 < x – 5 < 7

Example•Solve and graph the solution.•-3 < -2x + 1 ≤ 9

You Try!•Solve and graph the solution.•-1 ≤ 2x + 3 < 7

Solving with “or”•Solve each inequality separately.•Be sure to keep the “or” in your answer!•Example:•b + 7 < 2 or 2b < 8

Example•Solve and graph the solution.•3x- 5 < -8 or 2x – 1 > 5

You Try!•Solve and graph the solution.•4x + 1 ≤ -11 or 3x – 4 ≥ 5

Absolute Value Inequalities•Remember absolute value can be used for distance.•Example: |x| < 2 •Means “the distance between x and 0 is less than 2”

•Can be written -2 < x < 2

•|x| > 2•Means “the distance between x and 0 is greater than 2”

•Can be written x < -2 or x > 2

Solving Abs. Value•For less than rewrite as “and”

▫|ax + b| < c -c < ax + b < c

•For greater than rewrite as “or”

▫ |ax + b| > c ax + b < -c or ax + b > c

Example•Solve and graph:•|x + 7| ≤ 2

Example•Solve and graph:•|x – 3| ≥ 4

You Try!•Solve and graph:•|4x – 5| ≤ 11

•|2x + 5| ≥ 8

No Solution•Can an absolute value expression ever be negative?•NO!!•So, inequalities like:▫|x| < -1 ▫|x| ≤ -2 •have no solution.

All Real Numbers•When is an absolute value expression positive?•ALWAYS!•So, inequalities like:▫|x| > -1▫|x| ≥ -2 •have solution all real numbers.

You Try!•Tell if each inequality has no solution or all real numbers.

•|x| > -6

•|x – 7| < -1

•|3x + 9| ≥ -4

•|x + 6| < 0