Algebra 2

Chapter 1.6 Solving Compound & Absolute

Value Inequalities

Target Goals:1. Solve compound inequalities2. Solve absolute value inequalities

Compound Inequalities

• 2 inequalities joined by “and” or “or” (sometimes with words, sometimes with symbols)

• 2 Types:– “And” Compound Inequalities– “Or” Compound Inequalities

• You will always have 2 inequalities!!!

“And” Compound Inequalities

• The compound inequality is true if both of its equalities are true.

• When looking at the graph, this is where the shading intersects.

Ex 1) Solve

Method 1: Solve separately Method 2: Solve together

12 4 8 32x

12 4 8 32x 12 4 8 and 4 8 32x x 20 4x 5 x

5x

4 24x 6x

12 4 8 32x 20 4 24x

5 6x

and 5x 6x and

Ex 1) Graph the solution set on a number line and express the solution in interval notation.

5x and 6x

0 6-5

0 6-5

5x

6x

0 6-5

Interval Notation: [-5, 6)

“Or” Compound Inequalities

• The compound inequality is true if either of its equalities are true. Both don’t have to be true.

• When looking at the graph, this is wherever the shading naturally falls.

Ex 2) Solve 3 2 or 4x x

3 2 or 4x x

1x 4xor

1x -1 0 4

[

Interval Notation: (-∞, -1) or [4, ∞)

4x

“Solve.”

“Graph the solution set on a number line.”

“Express the solution in interval notation.”

Absolute Value Inequalities

Absolute value inequality Which compound inequality?

ax b c

ax b c

ax b c

ax b c

or ax b c ax b c

or ax b c ax b c

and ax b c ax b c

and ax b c ax b c

Great”or”

Less th”and”

Ex 3) Solve . Graph the solution set on a number line and express the solution in interval notation.• “and” or “or”?• 2 inequalities?

2d

AND

d < 2 and –d < 2d < 2 and d > -2

-2 0 2

)(

Interval Notation: (-2, 2)

Ex 4) Solve . Graph the solution set on a number line and express the solution in interval notation.

• “and” or “or”?• 2 inequalities?

3d

OR

d ≥ 3 or –d ≥ 3d ≥ 3 or d ≤ -3

-3 0 3[]

Interval Notation: (-∞, -3] or [3, ∞)

Ex 5) Solve . Graph the solution set on a number line and express the solution in interval notation.

• “and” or “or”?• 2 inequalities?

4 7 13x

OR

4x – 7 > 13 or -4x + 7 > 13

x > 5 or x < -3/2

4x > 20x > 5

-4x > 6x < -3/2

-3/2 0 5()

Interval Notation: (-∞, -3/2) or (5, ∞)

Ex 6) Solve . Graph the solution set on a number line and express the solution in interval notation.• “and” or “or”?• 2 inequalities?

5 2 17y

AND

5y + 2 ≤ 17 and –5y – 2 ≤ 17

y ≤ 3 and y ≥ -19/5

-19/5 0 3

][

Interval Notation: [-19/5, 3]

5y ≤ 15 y ≤ 3

-5y ≤ 19 y ≥ -19/5

Extention/Challenge

• What type of solution would you have if you have an “and” and no shading intersects on the graph?

No solution!

()

Extention/Challenge

• What type of solution would you have if you have an “or” and the entire line has been shaded?

Infinitely many solutions or all real numbers

][

Target Goals

1. Solve compound inequalities– (ex 1-6)

2. Solve absolute value inequalities– (ex 3-6)