Download ALG1 LT2.5 Compound Inequalities Notes

10/26/2015
Warmup: Solve and graph
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5𝑥 − 6 < 4
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−5 ≤ 3 4 − 𝑥
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3𝑥 − 2 ≤ 4𝑥 + 9
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7 𝑥 − 11 ≤ 3𝑥 + 4𝑥 − 3
Section 2.5
Compound Inequalities
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What is a combined compound
inequality?
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1) You take two inequalities, and combine
them into one. Like compound sentences.
Example.
5>𝑥
𝑎𝑛𝑑
𝑥 > −5
We can write this as a compound
inequality. Make sure that the signs face →
−5 < 𝑥 < 5
The graph is of the two original
inequalities on the same number line.
Example on next slide.
−5 < 𝑥 < 5
We graph it like it looks. 𝑥 is in between those two numbers!
The same “filled in circle” and “hollow circle” process applies!
𝑥 > −5
You Graph
5>𝑥
−6 ≤ 𝑥 ≤ 10
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Important:
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Compound Inequalities that are combined
always have the symbols facing the same
way, and generally facing right.
The variables will always be in the middle.
They go from small to large.
Examples. −21 ≤ 𝑥 < 23
.5 < 𝑥 ≤ 1
−4 ≤ 𝑥 ≤ 80
−2 ≤ 𝑥 − 6 < 8
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Are these written correctly?
1) 4 ≤ 𝑛 < −1
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2) 𝑓 < 45 < 99
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3) 19 > 𝑥 < 11
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4) 2 > 𝑥 > −3
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Oh heck no!
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Uncombined Compound
inequalities
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Compound Inequalities that are not
combined have an “or” between two
inequalities.
These should always go in opposite
directions on the graph.
Examples. x > 5 𝑜𝑟 𝑥 < −5
x > 4 𝑜𝑟 𝑥 ≤ −5
Graphing Uncombined Compound
inequalities
Examples 5 < 𝑥 𝑜𝑟 𝑥 ≤ −5
𝑥 ≤ −5
You Graph
6≤𝑥
5<𝑥
𝑜𝑟 2𝑥 ≤ −10
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Interpreting a graph.
We need to make sure we are able to
interpret a graph and find it’s inequality.
Example.
Solving compound Inequalities
The formal way.
Example 13 < 4 + 3𝑥 ≤ 55
STEP 1) write as 2 inequalities
13 < 4 + 3𝑥 𝑎𝑛𝑑
4 + 3𝑥 ≤ 55
STEP 2) Solve Each separately
13 < 4 + 3𝑥 𝑎𝑛𝑑
4 + 3𝑥 ≤ 55
9 < 3𝑥 𝑎𝑛𝑑
3𝑥 ≤ 51
3 < 𝑥 𝑎𝑛𝑑
𝑥 ≤ 17
 STEP 3) COMBINE AGAIN
3 < 𝑥 ≤ 17
You did it! Now that we’ve got that,
Let me show you a faster way!
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The Fast Way of solving…
To solve the compound inequality, we
have to do inverse properties to BOTH
sides.
 4 < 𝑥 − 3 < 13
+3
+3 +3
Now we add to BOTH
sides!
 7 < 𝑥 < 16
We are done!
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Watch out for flipping the sign. That still
applies!
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Example 2) Find the solutions and graph.
−5 ≤ 7 − 2𝑥 < 11
−7 −7
−7
Sub on all sides.
−12 ≤ −2𝑥 < 4
Divide all sides by
-2. FLIPS THAT
SIGN!
6 ≥ 𝑥 > −2
Then write it the
correct way
−2 < 𝑥 ≤ 6
One more Example with sign
flippage.
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Try this for yourself! #2 is beastly
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1) 22 ≤ 4 − 6𝑞 < 64
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2) 45 < 13 + 15𝑥 < 135
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P.101
AP 2.6 All
P.102
PS 2.6 #1-4
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Check Yoself
1) −10 < 𝑞 ≤ −3
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2) 2 15 < 𝑥 <815
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