Exercise Solve x 2 = 4. x = ± 2. Solve x 2 = – 4. no real solution Exercise.
Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 <...
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Transcript of Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 <...
![Page 1: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/1.jpg)
Warm UpSolve each inequality. 1. x + 3 ≤ 10
2.
5. 0 ≥ 3x + 3
4. 4x + 1 ≤ 25
x ≤ 7
23 < –2x + 3 –10 > x
Solve each inequality and graph the solutions.
x ≤ 6
–1 ≥ x
![Page 2: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/2.jpg)
SECTION 3.6
Solving Compound Inequalities
![Page 3: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/3.jpg)
5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
California Standards
![Page 4: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/4.jpg)
The inequalities you have seen so far are simple inequalities. When two simple inequalities are combined into one statement by the words AND or OR, the result is called a compound inequality.
Vocabulary
![Page 5: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/5.jpg)
REVIEWING INEQUALITIES SYMBOLS
≥
<
>
≤Less than or equal to
Greater than
Greater than or equal to
Less than
![Page 6: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/6.jpg)
IDENTIFY EACH SYMBOL
1) ≥ 2) <3) =4) ≤5) >6) ≠7) ≈
“is greater than or equal to”
“is less than”
“is equal to”
“ is less than or equal to”
“ is greater than”
“is approximately equal to”
“ is not equal to”
![Page 7: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/7.jpg)
![Page 8: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/8.jpg)
LET’S TRY ONE TOGETHER.Step 1: Make two equations
Step 2: undo addition or subtraction
Step 3: Solve
Step 4: Graph
All real numbers greater than or equal to two AND all real numbers less than six.
4 820 6
2 < x < 6
![Page 9: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/9.jpg)
LET’S TRY ONE TOGETHER. Step 1: Make two equations
Step 2: undo addition or subtraction
Step 3: undo division and multiplication
Step 4: Solve
Step 5: Graph
All real numbers greater than or equal to negative four AND all real numbers less than three.
0 4-2-4 2
22 2 2 -4 < x < 3
![Page 10: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/10.jpg)
Short MethodSolve the compound inequality and graph the solutions.
8 < 3x – 1 ≤ 11
8 < 3x – 1 ≤ 11+1 +1 +1
9 < 3x ≤ 12
3 < x ≤ 4
1. add 1 to each part of the inequality.
2. divide each part of the inequality by 3 to undo the multiplication.
The solution set is {x:3 < x ≤ 4}.
![Page 11: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/11.jpg)
–5–4 –3 –2–1 0 1 2 3 4 5
Graph 3 < x.
Graph x ≤ 4.
Graph the intersection by finding where the two graphs overlap.
3 < x ≤ 4
![Page 12: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/12.jpg)
LET’S TRY ONE TOGETHER. Step 1: It’s already two equations
Step 2: undo addition or subtraction
Step 3: Solve
Step 4: Graph
All real numbers greater than five OR all real numbers less than one.
3 71-1 5 a > 5 OR a < 1
![Page 13: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/13.jpg)
LET’S TRY ONE TOGETHER. Step 1: It’s already two equations
Step 2: undo addition or subtraction
Step 3: Solve
Step 4: Graph
All real numbers greater than five OR all real numbers less than two.
3 71-1 5
a < 2 OR a > 5
![Page 14: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/14.jpg)
MATCH THE COMPOUND INEQUALITY WITH THE CORRECT GRAPH
1. 0 < x + 2 < 5
2. -4 + a > 1 OR -4 + a < -3
3. -3 < x + 2 < 3
4. 2 < x + 2 < 5
5. x + 2 < -6 OR x + 2 > -2
2-2 0
40 2
51 3
0-4 -2
-4-8 -6
![Page 15: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/15.jpg)
NOW YOU TRY…SOLVE AND GRAPH THE COMPOUND INEQUALITY
1. -3 < x + 2 < 7
2. x – 1 < -1 OR x – 5 > -1
3. 2 < x + 2 < 5
4. 11 < 2x + 3 < 21
5. n + 2 < 3 OR n + 3 > 7
5-5 0
40 2
40 2
84 6
51 3
-5 < x < 5
x < 0 OR x > 4
x < 1 OR x > 4
0 < x < 3
4 < x < 9
![Page 16: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/16.jpg)
The compound inequality is x ≤ –8 OR x > 0.
Write the compound inequality shown by the graph.
![Page 17: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/17.jpg)
The compound inequality is –9 < x AND x < –2 (or –9 < x < –2).
Write the compound inequality shown by the graph.
![Page 18: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/18.jpg)
MATCH THE FOLLOWING
1. Inequality
2. Natural Numbers
3. Inverse Operations
4. Like Terms
5. Compound Inequality
A. the set of counting numbers
B. two inequalities that are combined into one statement by the word AND or OR.
C. terms that contain the same variable raised to the same power
D. a mathematical statement that compares two expressions by using one of the following signs: <, >, <, >, or ≠
E. operation that “undo” each other
![Page 19: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/19.jpg)
IN SUMMARY
Today you learned that two inequalities that are combined into one statement by the word AND or OR is called a compound inequality.
If it contains the word AND it is split into two equations and the graph is in between two points.
If it contains the word OR the graphs go in opposite directions from each point.
![Page 20: Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. 5. 0 ≥ 3 x + 3 4. 4 x + 1 ≤ 25 x ≤ 7 23 < –2 x + 3 –10 > x Solve each inequality and graph the solutions.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649d975503460f94a80a2b/html5/thumbnails/20.jpg)
Lesson Quiz
Solve each compound inequality and graph the solutions.
1. 2 ≤ 2w + 4 ≤ 12
–1 ≤ w ≤ 4
2. 3 + r > −2 OR 3 + r < −7
r > –5 OR r < –10
Write the compound inequality shown by each graph.
4. x < −7 OR x ≥ 0
5.−2 ≤ a < 4