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 < 60 You 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 > 2

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, B A

The numbers 4 and 6 are in both sets. 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 B A = {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 5 3 4 2 6 {x / x ≥ 3} [ 1 5 3 4 2 6 { x / 3 ≤ x ≤ 5 1 5 3 4 2 6 [ ]

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 < 4 4 8 6 7 5 9 ) 4 8 6 7 5 9 )

Example 2 The solution set is ( - ∞, 4) Step 3: Graph the compound inequality { x / x < 9 and x < 4} = { x / x < 4} 3 7 5 6 4 8 ) The solution set is ( - ∞, 4)

Solve: x + 5 < 9 and 3x -1 < 2 Give it a try!  Solve: x + 5 < 9 and 3x -1 < 2

Example 3 Solve 2 x ≥ 0 and 4 x – 1 ≤ -9 First 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 < 7 Step 1: Isolate the x in the middle 2 < 4 – x < 7 2 – 4 < 4 – x – 4 < 7 – 4 -2 < -x < 3 -2 < - x < 3 -1 -1 -1 2 > x > 3 Subtract 4 from all 3 parts Divide all three parts by -1 Must reverse symbol because dividing by a negative.

2 > x > -3 is equivalent to -3 < x < 2 Example 4 2 > x > -3 is equivalent to -3 < x < 2 Interval Notation (-3, 2) Graph the inequality -3 1 -1 -2 2 ( )

Give it a try!  Solve: 5 < 1 – x < 9

Example 5 Solve

Example 5 Graph the solution Interval Notation

Give it a try! 

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 B A: {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 ≥ 5 First 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 < 0 Step 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