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Lesson Menu Five-Minute Check (over Lesson 5–3) CCSS Then/Now New Vocabulary Example 1:Solve and Graph an Intersection Example 2:Real-World Example: Write and Graph a Compound Inequality Example 3:Solve and Graph a Union

Over Lesson 5–3 5-Minute Check 1 A.{x | x < 30} B.{x | x < 20} C.{x | x < 15} D.{x | x < 10} Solve 3x – 15 < 45.

Over Lesson 5–3 5-Minute Check 2 A.{p | p  18} B.{p | p  –18} C.{p | p  18} D.{p | p  –18} Solve 2p – 22  4p + 14.

Over Lesson 5–3 5-Minute Check 3 A.x < 64 B.x < 4 C.x > –64 D.x > –4 Solve –3 – < 13. __ x 4

Over Lesson 5–3 5-Minute Check 4 A.2(n – 5 ) > 7 B.2(n – 5 ) < 7 C.2 < (n – 5 )7 D.2n < 7 – 5 Choose the correct inequality for two times the difference of a number and five is greater than seven.

Over Lesson 5–3 5-Minute Check 5 A.k  –5 B.k  5 C.k  7 D.k  7 Solve 8(6 – k) + 2k  –15 – (–3k).

CCSS Content Standards A.CED.1 Create equations and inequalities in one variable and use them to solve problems. A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Mathematical Practices 1 Make sense of problems and persevere in solving them. 8 Look for and express regularity in repeated reasoning. Common Core State Standards © Copyright National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

Then/Now You solved absolute value equations with two cases. Solve compound inequalities containing the word and, and graph their solution set. Solve compound inequalities containing the word or, and graph their solution set.

Vocabulary compound inequality – two or more inequalities that are connected by the words and or or Intersection – graph where the two inequalities overlap Union – the graph of a compound inequality containing or; the solution is a solution of either inequality not necessarily both

Example 1 Solve and Graph an Intersection Solve 7 < z + 2 ≤ 11. Graph the solution set. First express 7 < z + 2 ≤ 11 using and. Then solve each inequality. 7 < z + 2 and z + 2 ≤ 11 Write the inequalities. 7 – 2 < z + 2 – 2 z – ≤ 11 – 2 Subtract 2 from each side. 5< z z ≤ 9 Simplify. The solution set is {z | 5 < z ≤ 9}.

When considered together, two inequalities such as h ≥ 52 and h ≤ 72 form a compound inequality. A compound inequality containing and is only true if both inequalities are true. Its graph is where the graphs of the two inequalities overlap. This is called the intersection of the two graphs. The intersection can be found by graphing each inequality and then determining where the graphs intersect.

x ≥ 3 x < 7 x ≥ 3 and x < 7 The statement 3 ≤ x < 7 can be read as x is greater than or equal to 3 and less than 7 or x is between 3 and 7 including 3

Example 1 Graph z ≤ 9. Find the intersection. Graph 5 5. Answer: Solve and Graph an Intersection

Solve -2 ≤ x – 3 < 4 First, express -2 ≤ x – 3 < 4 using and. Then solve each inequality. -2 ≤ x – 3 and x – 3 < ≤ x x < 7 The solution set is {x│1 ≤ x < 7}

Now graph x ≥ 1 X < 7 1 ≤ x < 7

Solve each compound inequality. Then graph the solution set. 1)y – 3 ≥ -11 and y – 3 ≤ -8 {y│-8 ≤ y ≤ -5} 2) 6 ≤ r + 7 < 10 { r│-1 ≤ r < 3}

Example 1 Solve –3 < x – 2 < 5. Then graph the solution set. A.{x | –1 < x < 7} B.{x | –5 < x < 3} C.{x | x < 7} D.{x | –1 < x < 3}

Another type of compound inequality contains the word or. A compound inequality containing or is true if at least one of the inequalities is true. Its graph is the union of the graphs of two inequalities. x > 2 x ≤ -1 x > 2 or x ≤ -1

When solving problems involving inequalities within is meant to be inclusive, so use ≤ or ≥. Between is meant to be exclusive, so use.

Example 2 Write and Graph a Compound Inequality TRAVEL A ski resort has several types of hotel rooms and several types of cabins. The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount that a guest would pay per night at the resort.

Example 2 Write and Graph a Compound Inequality Graph n ≤ 89. Find the union. Answer:{n | n ≤ 89 or n ≥ 109} Now graph the solution set. Graph n ≥ 109.

Example 2 TICKET SALES A professional hockey arena has seats available in the Lower Bowl level that cost at most $65 per seat. The arena also has seats available at the Club Level and above that cost at least $80 per seat. Write and graph a compound inequality that describes the amount a spectator would pay for a seat at the hockey game. A.c ≤ 65 or c ≥ 80 B.c ≥ 65 or c ≤ 80 C.c ≥ 65 or c ≥ 80 D.c ≤ 65 or c ≤ 80

Example 3 Solve and Graph a Union Solve 4k – 7 ≤ 25 or 12 – 9k ≥ 30. Graph the solution set. or

Example 3 Solve and Graph a Union Graph k ≤ 8. Graph k ≤ –2. Answer: Notice that the graph of k ≤ 8 contains every point in the graph of k ≤ –2. So, the union is the graph of k ≤ 8. The solution set is {k | k ≤ 8}. Find the union.

Solve -2m + 7 ≤ 13 or 5m + 12 > 37. Then graph the solution set. -2m + 7 ≤ 13 or 5m + 12 > m ≤ 6 5m > m ≥ -3 m > 5

m ≥ -3 m > 5 m ≥ -3 or m > 5 Notice that the graph of m ≥ -3 contains every point in the graph of m > 5. So the union is the graph of m ≥ -3. The solution set is {m│m ≥ -3}

Solve each compound inequality. Then graph the solution set. 1)a + 1 < 4 or a – 1 ≥ 3 {a│a < 3 or a ≥ 4} 2) x ≤ 9 or 2 + 4x < 10 { x│x ≤ 9}

Example 3 Solve –2x Then graph the solution set. A.{x | x > 1} B.{x | x < –5} C.{x | x > –5} D.{x | x < 1}

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