Chapter 2.5 – Compound Inequalities

Objectives

I will find the intersection of two sets.I will solve compound inequalities

containing and.I will find the union of two sets.I will solve compound inequalities

containing or.

What’s the difference?

You get a discount if you are at least 18 years old and no more than 60 years old.

18 < x < 60You get a discount if you are less than

18 years old or at least 60 years old.

18 > x > 60

Compound Inequalities

Two inequalities joined by the words and or or are called compound inequalities.

Compound Inequalities

x + 3 < 8 and x > 225 10 7

3x

or x

Intersection of Two Sets

The intersection of two sets. A and B, is the set of all elements common to both sets. A intersect B is denoted by,

A B

A B

Example 1

If A = { x / x is an even number greater than 0 and less than 10} and B = {3, 4, 5, 6} find A ∩ B.

List the elements in sets A and BA = {2, 4, 6, 8}B = {3, 4, 5, 6}

The numbers 4 and 6 are in both sets.The intersection is {4, 6}

You try it!

Find the intersection:

{1, 2, 3, 4, 5} ∩ {3,4,5,6}

{3,4,5}

Solutions to compound inequalities

A value is a solution of a compound inequality formed by the word and if it is a solution of both inequalities.

For example, the solution set of the compound inequality x ≤ 5 and x ≥ 3 contains all values of x that make both inequalities true.

Compound Inequalities

A compound inequality such as x ≥ 3 and x ≤ 5 can be written more compactly.

3 ≤ x ≤ 5

Graphing compound inequalities

{ x / x ≤ 5}

1 53 42 6

]

1 53 42 6

{x / x ≥ 3}[

1 53 42 6

{ x / 3 ≤ x ≤ 5

[ ]

Example 2

Solve x – 7 < 2 and 2x + 1 < 9

Step 1: Solve each inequality separately

x – 7 < 2 and 2 x + 1 < 9

x < 9 and 2x < 8

x < 9 and x < 4

Example 2

Step 2: Graph the two intervals on two number lines to find their intersection.

x < 9

x < 44 86 75 9

)

4 86 75 9

)

Example 2

Step 3: Graph the compound inequality

{ x / x < 9 and x < 4} = { x / x < 4}

3 75 64 8

)

The solution set is ( - ∞, 4)

Give it a try!

Solve: x + 5 < 9 and 3x -1 < 2

Example 3

Solve 2 x ≥ 0 and 4 x – 1 ≤ -9First Step: Solve each inequality

separately.

Example 3

Step 2: Graph the intervals to find their intersection.

Example 3

Step 3: Graph the intersection

You try it!

Solve: 4x ≥ 0 and 2x + 4 ≤ 2

Example 4

Solve 2 < 4 – x < 7Step 1: Isolate the x in the middle

2 < 4 – x < 72 – 4 < 4 – x – 4 < 7 – 4

-2 < -x < 3-2 < - x < 3 -1 -1 -12 > x > 3

Subtract 4 from all 3 parts

Divide all three partsby -1

Must reverse symbolbecause dividing by a negative.

Example 4

2 > x > -3 is equivalent to -3 < x < 2

Interval Notation (-3, 2)

Graph the inequality

-3 1-1 0-2 2

( )

Give it a try!

Solve: 5 < 1 – x < 9

Example 5

Solve 21 5 2

3x

99

2x

Example 5

Graph the solution

Interval Notation

Give it a try!

3 1 52x

Union of two sets

The solution set of a compound inequality formed by the word or is the union of the solution set of two inequalities.

The union of two sets, A and B, is the set of elements that belong to either of the sets. A union B is denoted by

A U B

Example 6

If A = {x / x is an even number greater than 0 and less than 10} and B = {3, 4, 5, 6} Find A U B.

List the elements in Set A and Set BA: {2, 4, 6, 8}

B: {3, 4, 5, 6}

Union : {2,3, 4, 5, 6, 8}

Union

A value is a solution of a compound inequality formed by the word or if it is a solution of either inequality.

Graph of {x / x ≤ 1}

Graph of { x / x ≥ 3}

Unions

Graph of { x/ x ≤ 1 or x ≥ 3}

Give it a try!

Find the union:

{1, 2, 3, 4, 5} U {3, 4, 5, 6}

{1,2,3,4,5,6}

Give it a try!

Solve: 5 x – 3 ≤ 10 or x + 1 ≥ 5First solve each inequality separately.

Example 7

Now we can graph each interval and find their union.

Example 7:

Solution set:

Give it a try!

Solve: 3x – 2 ≥ 10 or x – 6 ≤ -4

Example 8

Solve: – 2x – 5 < -3 or 6x < 0Step 1: Solve each inequality separately

Example 8

Graph each interval and find their union

Give it a try!

Solve: x – 7 ≤ -1 or 2x – 6 ≥ 2