Solving Absolute-Value, Compound, and Quadratic Inequalities

Reminder: Compound Inequalities

The following are examples of how to algebraically write the following graphs:

0 4

0≤x<4

-1 2

x<-1 or x>2

REMINDER

How to solve a one variable inequality.

Solving a 1 Variable Inequality

2 7 12x

2 7 12x

2 5x 2.5x

0

x

x = 0 x = 3

7 < 12 13 < 12True False

Find the Boundary Test Every Region

2.5x

Represent the solutions to the following inequality algebraically and on a number line.

2 3 7 12 2 0 7 12

Change inequality to equality

Solve

Plot Boundary Point(s)

Pick a point in each region

Substitute into Original

Shade True Region(s)

Algebraic Solution

Closed or Open Dot(s)?

Graphical Solution

Apply this method to more complicated Ineqaulities

Solving an Absolute Value Inequality

2 3x

2 3x 2 3x

2 3 2 3x x

5 or 1x x

0

x

x = -2 x = 0 x = 6

4 > 3 2 > 3 4 > 3True False True

Find the Boundary Test Every Region

1 5x or x

Represent the solutions to the following inequality algebraically and on a number line.

2 2 3 6 2 3 0 2 3

Change inequality to equality

Solve

Plot Boundary Point(s)

Pick a point in each region

Substitute into Original

Shade True Region(s)

Algebraic Solution

Closed or Open Dot(s)?

Graphical Solution

Solving a Compound Inequality

12 2 8 2x

12 2 8x

2 or 3x x

0

x

x = -3 x = 0 x = 4

-12<-14≤-2 -12<-8≤-2 -12<0≤-2False True False

Find the Boundary Test Every Region

2 3x

Represent the solutions to the following inequality algebraically and on a number line.

12 2 3 8 2 12 2 4 8 2 12 2 0 8 2

Change inequality to two equalities

Solve Both

Plot Boundary Point(s)

Pick a point in each region

Substitute into Original

Shade True Region(s) Algebraic

Solution

Closed or Open Dot(s)?

Graphical Solution

2 8 2x

4 2x 2 6x

Solving a Quadratic Inequality

2 2 1 4x x

2 2 1 4x x 2 2 3 0x x

3 1 0x x

3 or 1x x

0

x

x = -4 x = 0 x = 2

9 < 4 1 < 4 9 ≤ 4False True False

Find the Boundary Test Every Region

3 1x

Represent the solutions to the following inequality algebraically and on a number line.

24 2 4 1 4 2

2 2 2 1 4 20 2 0 1 4

Change inequality to equality

Solve

Plot Boundary Point(s)

Pick a point in each region

Substitute into Original

Shade True Region(s) Algebraic

Solution

Closed or Open Dot(s)?

Graphical Solution